GRADE 8 | UNIT 3 Linear Equations

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STUDY GUIDE

GRADE 8 | UNIT 3

Linear Equations

Table of Contents

Introduction .................................................................................................................................... 3

Test Your Prerequisite Skills ........................................................................................................ 4

Objectives ...................................................................................................................................... 5

Lesson 1: The Rectangular Coordinate System

- Warm Up! ........................................................................................................................... 5

- Learn about It! ................................................................................................................... 6

- Let’s Practice! ..................................................................................................................... 8

- Check Your Understanding! ............................................................................................ 13

Lesson 2: Linear Equations in Two Variables

- Warm Up! ......................................................................................................................... 15

- Learn about It! ................................................................................................................. 16

- Let’s Practice! ................................................................................................................... 16


Lesson 3: Slope of a Line

- Warm Up! ......................................................................................................................... 21

- Learn about It! ................................................................................................................. 22

- Let’s Practice! ................................................................................................................... 26


Lesson 4: Writing Linear Equations in Standard and Slope-Intercept Form

- Warm Up! ......................................................................................................................... 34

- Learn about It! ................................................................................................................. 35

- Let’s Practice! ................................................................................................................... 36



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STUDY GUIDE

Lesson 5: Graphing Linear Equations

- Warm Up! ......................................................................................................................... 42

- Learn about It! ................................................................................................................. 43

- Let’s Practice! ................................................................................................................... 45


Challenge Yourself! ..................................................................................................................... 54

Performance Task ....................................................................................................................... 54

Wrap-up ....................................................................................................................................... 56

Key to Let’s Practice! .................................................................................................................... 58

References ................................................................................................................................... 61


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STUDY GUIDE

GRADE 8 | MATHEMATICS

UNIT 3

Linear Equations

Imagine hearing your father telling your mother, “Our

profit dropped 1% last April, about 2% last month and

more so this month. We have to adjust our prices to

sustain the operating costs of our store.” How can you help

your parents do the necessary adjustments? Knowledge

about linear equations will enable you to somehow alleviate their burdens. You can

even use it in helping compute for the monthly budget of your family.

Linear equations are a powerful tool to determine how

much profit you can obtain in any business venture.

Companies may use linear equations in projecting sales

and profits and eventually predicting the future of their

business.

Linear equations have extensive real life applications. They may be

used to find the distance traveled by cars, planes, and boats, or

even the time it takes to travel a certain distance when these modes

of transportation are travelling at a constant speed.

You need not go far; linear equations can help you compute how much you should score

in the remaining quizzes to obtain a desired average.

This unit shall focus on linear equations, its applications, and how it can be represented

graphically in the rectangular coordinate system.

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Table of Contents


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STUDY GUIDE

Before proceeding to the subject matter, be sure that you have mastered the following

skills that will help you understand the topic better. In a separate sheet of paper, answer

the following questions as honestly as possible. Feel free to review your past lessons if

need arises.

1. Evaluate the following algebraic expressions given the values of the variables:

a. , where

b. , where

c.

, where , , ,

2. Simplify the following rational expressions

a.

b.

c.

3. Find the solution of a given equation

a.

b.

c.

Evaluating algebraic expressions for given values of the variables

Simplifying coefficients in fraction form

Finding the solution of a given equation

Test Your Prerequisite Skills


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STUDY GUIDE

At the end of this unit, you should be able to

describe the rectangular coordinate system;

identify linear equations in two variables;

differentiate between linear equations with one variable and two variables;

calculate the slope of a line given two points;

identify the slope of a line in a graph;

write linear equations in standard form and in slope-intercept form; and

sketch the graph of a linear equation.

Where in the World is Lumen San Diego?

Materials Needed: world map, marker

Instructions:

1. This activity may be done individually or in pairs.

2. Using the world map, draw a vertical line along the Greenwich Meridian.

(The Greenwich Meridian is also known as the prime meridian is the imaginary

line passing through Greenwich, England that is at 0° longitude.)

3. Similarly, draw a horizontal line along the equator (0° latitude).

4. Use the intersection of the two lines as the point of origin.

5. Each grid in the map is used as one unit.

6. Locate Lumen San Diego using the following directions:

Lesson 1: The Rectangular Coordinate System

Objectives

Warm Up!


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STUDY GUIDE

a. From the origin, Lumen San Diego travelled 5 units to the right and two units

up to the second most populated country in the world. What is this country?

b. She then went two units left and one unit up to a place where majority of the

historically known cradle of civilization (Mesopotamia) was. In which modern

day country can you find most parts of Mesopotamia?

c. Her last stop was 3 units to the left and 1 unit up to the country famed for its

fashion, classical arts, museums and monuments.

7. Where was Lumen San Diego’s last stop? Name the country.

The grid of the lines of the map you used in the Warm Up! activity is like the rectangular

coordinate system.

This is a number line.

This is a rectangular coordinate plane.

Learn about It!


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STUDY GUIDE

Notice that the x-axis and the y-axis intersect at a point and that these number lines

divide the plane into four regions.

Definition 1.3: The origin is the point where the x-axis

and the y-axis in the rectangular

coordinate plane intersect.

Definition 1.3: The y-axis is the vertical number line in

the rectangular coordinate plane.

Definition 1.2: The x-axis is the horizontal number line

in the rectangular coordinate plane.

Definition 1.1: The rectangular coordinate plane

is composed of two number lines

perpendicular to each other.


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STUDY GUIDE

Each point on the rectangular coordinate plane is represented by the coordinates of and

, called an ordered pair, .

Example 1: Plot the point on the rectangular coordinate plane and identify in which

quadrant it is located.

Solution: Given the point , and . The point is located in the first

quadrant.

Definition 1.4: The quadrants are the four regions into

which the intersecting x-axis and y-axis

divide the rectangular coordinate plane.

Let’s Practice!


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STUDY GUIDE

Try It Yourself!

Plot the point on the rectangular coordinate plane. On what quadrant is it

located?

Example 2: What ordered pair describes the given point on the coordinate plane shown?


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STUDY GUIDE

Solution: Looking at the point, we see that it is aligned with and . Thus,

the point is represented by .

Try It Yourself!

What ordered pair describes the given point on the coordinate shown? In which

quadrant does it lie?

Example 3: The three points marked are vertices of a rectangle. Find the fourth vertex.


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STUDY GUIDE

Solution: Connect the points to form a rectangle and determine the coordinates of the

fourth vertex.

Thus, the fourth vertex is at .

Try It Yourself!

The vertices of a triangle lie on points and . Plot the points and

draw the triangle.


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STUDY GUIDE

Real-World Problems

Example 4: Kilometer Zero in the Philippines is

located across the famous

monument of Jose Rizal in Luneta

Park, and is used as a distance

marker to any point in the

Philippines. Using Kilometer Zero

as the origin, determine the

ordered pairs for the locations of

Intramuros and Philippine General

Hospital.

Solution: If each grid line represents one unit, Intramuros is located two units left and

5 units up from the origin. Thus, the ordered pair for Intramuros is

From Kilometer Zero or the origin, Philippine General Hospital is located 6

units to the right and 3 units down. Thus, the ordered pair for Philippine

General Hospital is .

Example 5: From the Warm Up! activity, determine the coordinates of the places where

Lumen San Diego travelled.

Solution: First stop of Lumen was India which has coordinates . Her second stop

was Iraq with coordinates . Her last was France with coordinates ( .


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STUDY GUIDE

Try It Yourself!

Mario’s city was built similar to a rectangular coordinate plane. Using Mario’s house

as origin, the church is located at the point . The mall is located at the point

. If each unit represents 100 meters, how far is the church from the mall?

1. Identify the quadrant in which the following points lie:

a.

b.

c.

d.

Check Your Understanding!


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STUDY GUIDE

2. Plot the following points on the coordinate plane:

a.

b.

c.

d.

3. What ordered pair describes the point on the coordinate plane below?

4. The three points marked are vertices of a square. Find the fourth vertex.


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STUDY GUIDE

5. Two branches of a famous mall, and are connected by a train that has a station at

point ( of the coordinate plane. Mall A lies on the point and Mall B lies on

the point . Which of the two malls is closer to the train station?

Enjoy the Bus Ride!

Materials Needed: pen and paper

Instructions:

1. This activity is to be done individually.

2. The minimum fare for an ordinary bus is ₱10 and ₱12 for an air-conditioned bus.

3. Complete the table below by computing for the total amount of the fare for a

given number of passengers or for the missing number of passengers for a given

fare. Assume each passenger paid the minimum fare.

Number of

Passengers

Total Fare Amount in

Ordinary Bus

Total Fare Amount in Air-

conditioned Bus

1

2

₱40

₱84

8

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Lesson 2: Linear Equations in Two Variables

Warm Up!


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STUDY GUIDE

Variables are helpful in representing unknown values in a problem. As variables are

combined with numbers, they become expressions and represent more complex

quantities. Expressions that are equal to each other become equations.

You are already familiar with equations involving one variable, like , and

know how to solve it and get .

In the Warm Up! activity, you were acquainted with the use of two variables, one for the

number of passengers and the other for the fare. The corresponding values for the fare

depended on how much the minimum fare was.

A linear equation is called such because this type of equation forms a straight line when

graphed on the rectangular coordinate plane.

Such linear equations can always be expressed with the standard form, ,

where and are real numbers and and are not both equal to 0. The two variables

are and

Example 1: Are the equations and linear equations in two

variables?

Definition 2.1: A linear equation in two variables is an

equation with two variables, each of which

is having an exponent of one.

Let’s Practice!

Learn about It!


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STUDY GUIDE

Solution: The equation is a linear equation in two variables because it

has two variables and and each of which has an exponent or degree of 1.

The equation is not a linear equation in two variables. Though

it contains two variables, and , the degree of is 2 and not 1.

Try It Yourself!

Which of the following equations in two variables is/are linear?

a) b) c)

Example 2: In the equation , solve for when .

Solution:

Step 1: Substitute in the given equation.

Step 2: Solve for .

Thus, when .

Try It Yourself!

Given the equation . Solve for , when .


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STUDY GUIDE

Example 3: Is the ordered pair a solution of

Solution: To say that is a solution is to say that the x- and y- coordinates of the

ordered pair satisfy the given equation.

Step 1: Determine the x- and y- coordinates.

From , we know that and .

Step 2: Substitute the values of and into the equation.

Step 3: Verify if the answer is reasonable.

Since the left-hand side and the right-hand side of the equation are both

equal to 10, the ordered pair is a solution to the linear equation

.

Try It Yourself!

Which of the following ordered pairs is a solution to the equation ?

a) b) c)


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STUDY GUIDE

Real-World Problems

Example 4: Mrs. Ganzon wants her ₱1000.00 bill changed

into some ₱20 and ₱50 bills. If is the number

of ₱20 bills and is the number of ₱50,

translate the problem into a linear equation in

two variables.

Solution: Since is the number of ₱20 bills, the total amount of all the ₱20 bills is given

by .

Since is the number of ₱50 bills, the total amount of all the ₱50 bills is

given by .

The total amount should be equal to ₱1000. Thus, the desired equation is

.

Example 5: The sum of Lina’s age and her father’s age is 45. The difference of their ages

is 27. Write the two linear equations in two variables that model the problem.

Let Lina’s age be and her father’s age be .

Solution: The first statement says their ages have a sum of 45. Therefore, we have

The second statement says their ages have a difference of 27. Therefore, we

have

Note that the father has to be older than Lina, thus is assigned as the

minuend and is assigned as the subtrahend in the second equation.


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STUDY GUIDE

Try It Yourself!

Mr. San Pedro paid ₱360 for 6 adult’s tickets and 7

children’s tickets for the school play. Write a linear

equation for the given situation with the cost of the tickets

unknown. Let be the cost of an adult’s ticket and be the

cost of a child’s ticket.

1. In the equation , solve for when

a.

b.

c.

2. Are the equations and linear equations in two variables? Why

or why not?

3. Is the ordered pair a solution of ? Why?

4. The cost of a dozen apples and a dozen bananas is ₱250. If the cost of a single

apple is and the cost of a single banana is , write a linear equation in two

variables for the given information.

5. A certain number of 100-peso bills and 200-peso bills amount to ₱30 000. Write

an equation to represent this condition.



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STUDY GUIDE

You Raise Me Up!

Materials Needed: measuring tape or meter stick, pen, paper

Instructions:

1. This activity will be done in pairs.

2. You and your partner are to find a staircase inside your school premises and you

are to measure the steps before one landing.

3. Below are basic terminologies of some parts of the stairs:

a. Riser – vertical section of a step

b. Tread – horizontal section of a step

c. Landing – a level platform installed at a point where stairs change direction

4. Using varying number of steps, measure the total height of the risers of the steps

you are considering, as well as the total width of the treads.

5. Record your measurement on the table below.

Names: Location of Stairs:

No. of

Steps

Total Height of

the Risers

Total Width of the

Treads

Ratio of Total Height

of Risers to Total

Width of the Treads

1

2

3

4

5

Lesson 3: Slope of a Line

Warm Up!


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STUDY GUIDE

Every line is characterized by a slope. The slope gives us an idea of how steep the line is.

Looking at the mountain in the figure, the person at the foot of it can say it is very steep. A

person near a hill would describe the hill as less steep than the mountain. Lastly, a person

on a plain would say the plain is not steep at all.

If we use a line to model the sides of the mountain, the hill and the plain, we can describe

how steep it is using the value of the slope.

However, to be exact, the slope of a line is defined as the following:

The variable frequently used to denote the slope is .

In the Warm Up! activity, the total height of the risers may be thought of as the rise and

the total width of the treads may be thought of as the run.

Definition 3.1: The slope of a line is the ratio of the

change in vertical distance (rise) to the

change in horizontal distance (run).

Learn about It!


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STUDY GUIDE

This concept can be illustrated by the following formula.

There are three ways to find the slope depending on the information given. We will

concentrate on two of these in this lesson. The third way shall be presented in the next

lesson.

Given a graph

When you are given a graph of a line, find the slope of the line by using the formula

This may be computed when you have chosen two points on the line. For instance, to

calculate the slope of the following line, the following steps would be helpful:


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STUDY GUIDE

Step 1: Find the rise value.

Start at the lower point and count upwards until you reach the line where the

second point is.

We count 4 units up, thus rise = 4.

Step 2: Find the run value.


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STUDY GUIDE

From where you stopped in Step 1, resume counting towards the second

point (left or right). Counting to the left is assigned a negative sign, while

counting to the right is assigned a positive sign.

We count 3 units to the left, thus run = .

Step 3: Substitute the values obtained in the previous steps to the formula for the

slope.

Given two points

When you are given the coordinates of two points and , find the slope

of the line using the formula

.

For example, what is the slope of the line that passes through the points and

?

Step 1: Identify the value of the variables.


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STUDY GUIDE

You can freely assign the names and to the given points. For this

problem, let us use and .

Since and , from point , we have and .

Similarly, from point , we have and .

Step 2: Substitute the values into the formula.

Step 3: Simplify.

It is also important to note that two or more lines that have the same slope are parallel to

one another. Also, if the slope of one line is the negative reciprocal of the other, that is if

and , then the two lines are perpendicular.

Example 1: Find the slope of the given line.

Let’s Practice!


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STUDY GUIDE

Solution: We randomly choose a point on the line and trace the steps vertically and

horizontally until we get to another random point on the line.

Step 1: Find the rise value.

Start at the lower point and count upwards until you reach the level where

the second point is.

We count 2 units up. Thus, rise = 2.

Step 2: Find the run value.

From where you stopped in step 1, resume counting towards the second

point.


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STUDY GUIDE

We count 3 units to the left. Thus, run = –3.

Step 3: Identify the slope.

Try It Yourself!

Find the slope of the given line.

(Assume that one grid is one unit.)


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STUDY GUIDE

Example 2: What is the slope of the line that passes through the points and

?

Solution:

Step 1: Identify the values of the variables.

Let and . Thus, , , , and .

Step 2: Substitute the values into the formula.

Step 3: Simplify.

Thus, the slope of the line is 7.

Try It Yourself!

Find the slope of the line that passes through the points and .

Example 3: Which has a steeper slope: a line that passes through and or a

line that contains the points and ?

Solution:

Step 1: Find the slope of the first line passing through points and .


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STUDY GUIDE

Step 2: Find the slope of the second line with points and .

Step 3: Compare the two slopes.

Since

, the line containing the points and is steeper

than the other line.

Try It Yourself!

Compare the line containing and and the line that passes through

the points and . Which line is steeper? How are the lines related to each

other?

Real-World Problems

Example 4: Mrs. Cruz wants to bake a cake for her son’s birthday.

She observed the following readings: the temperature

of the oven after two minutes is and after 4

minutes. Find the rate of change of the temperature.

Solution: To find the rate of temperature change, we may use the concept of slope.

Step 1: Identify the ordered pairs.

Let denote the time and denote the temperature.


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STUDY GUIDE

When the time is 2 minutes, the temperature is . We may denote this

using the ordered pair .

When the time is 4 minutes, the temperature is . We may denote this

using the ordered pair .

Step 2: Find the slope.

From the formula of slope, we have

Thus, the rate of change of the temperature is 25°C/minute.

Example 5: As Marina eats frequently in her favorite restaurant, she noticed that the

time it takes the food to be served depends on how many people she is

ordering for. If she orders for two people, their food gets served in about 10

minutes. If she goes with her family, the six of them will have to wait for 20

minutes to get served. What is the rate at which people are served their

food?

Solution: Similar to Example 4, we may use the concept of slope.

Step 1: Identify the ordered pairs.

Let denote the number of people and denote the time.


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STUDY GUIDE

When Marina orders for 2 people, they get served after 10 minutes. We may

denote this using the ordered pair .

When Marina orders for 6 people, they get served after 20 minutes. We may

denote this using the ordered pair .

Step 2: Find the slope.

From the formula of slope, we have

The answer

may be interpreted as taking 5 minutes to serve to 2 people. The answer 2.5

may be interpreted as 2.5 minutes is required to serve one person.

Try It Yourself!

From the given table below, find the slope of the linear function and interpret what

the slope represents in the situation.

Daily Wage In a Factory

Amount of Time Worked

(in hours)

Compensation

(in pesos)

4 240

8 480

12 720


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STUDY GUIDE

1. Find the slope of the graph below.

2. What is the slope of the line that passes through the points and ?

3. Line passes through the points and while line passes though

and . What are the slopes of the two lines and how are they related to one

another?

4. Stephen walks in an inclined plane from one point to another (see figure below).

Find the steepness of the plane.



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STUDY GUIDE

5. A ladder touches a house 150 inches above the ground. Its base is 52 inches from

the house. What is the slope of the ladder? Assume the slope is positive.

Rice All You Can!

Materials Needed: pen and paper

Instructions:

1. This activity is to be done individually.

2. Your family is in the rice business and is selling both NFA and premium rice. NFA

rice costs ₱500 per sack of 50 kilos while premium rice costs ₱2000 per sack.

Complete the table below using the given information.

Sack of NFA

rice sold Cost

Sack of

Premium rice

sold

Cost Total cost

1 1

2 2

3 3

4 4

5 5

Lesson 4: Writing Linear Equations in Standard and Slope-

Intercept Form

Warm Up!


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STUDY GUIDE

In the Warm Up! activity, it can be seen that the number of sacks of rice sold and the cost

of the sacks form a linear relationship. From the information above, we can write a linear

equation with two variables for each type of sack.

Linear equations can be written in different forms. Each of these forms is useful

depending on what we know or want to know about a line.

The following are the most frequently used forms of linear equations.

1. Standard form

2. Slope-intercept form

3. Point-slope form

This topic will focus on the standard and slope-intercept forms.

Definition 4.1: The standard form of a linear equation

in two variables and , is an equation in

the form

,

where and are integers, and

and are not both equal to 0. By

convention, should also be

nonnegative.

Learn about It!


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STUDY GUIDE

The slope-intercept form is especially helpful when looking for the slope and y-intercept of

a linear equation. Also, the slope-intercept form is useful when the variable acts like a

dependent variable and the variable acts like an independent variable (i.e., when the

value of depends on the value of .)

Linear equations can be converted between the standard and slope-intercept forms. This

can be done by algebraic manipulation.

Converting Linear Equations from Standard Form to Slope-Intercept Form

To convert a linear equation from standard form to slope-intercept form, solve for y in

terms of x.

Converting Linear Equations from Slope-Intercept Form to Standard Form

The goal is to put the x and y variables on one side of the equation and the constant on

the other side. In the process, you should get rid of fractions or decimals to make the

coefficients in integer form and keep in mind that A should be non-negative.

Example 1: Convert to slope-intercept form.

Definition 4.2: The slope-intercept form of a linear

equation in two variables and is an

equation in the form

where is the slope and is the y-

intercept.

Let’s Practice!


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STUDY GUIDE

Solution: We solve for in terms of .

Subtract from both sides of the equation.

Rearrange the right-hand side so that the x-term comes first.

Divide both sides of the equation by 3.

Try It Yourself!

Write the equation into slope-intercept form.

Example 2: Convert to standard form.

Solution: Put the terms with and on one side of the equation, making sure that

the -term comes first.


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STUDY GUIDE

Get rid of fractions (if any) by finding the LCD and multiplying it to all the

terms in the equation. In this example, the only denominator is 5.

Make the coefficient of positive (if it is negative). Multiply all the terms

by −1 to do this.

Try It Yourself!

Write the equation in standard form.

Example 3: Is the equation the standard form of the equation ?

Solution: Convert to slope intercept form.

Thus, the equation is the standard form of the equation

.

Try It Yourself!

Find out if the equation is the slope-intercept form of .


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STUDY GUIDE

Real-World Problems

Example 4: You want to buy a present for your parent’s wedding

anniversary and you decided to save ₱200 a month from

your allowance. You have an initial savings of ₱400. How

much will you save in 3 months?

Solution:

Step 1: Identify a suitable equation form to be used.

Since we are given the amount saved each month (which may be

mathematically translated as the slope), and the initial savings (which may be

translated as the y-intercept), it is best to write an equation in slope-intercept

form to find the answer.

Step 2: Set up the equation.

Let be the number of months.

Let be the total amount of money you can save in months.

Since the amount saved each month is indicated, ₱200, we shall assign this

as the slope or . We can think of it as the rate of change of the amount of

savings per month. The initial saving is like the y-intercept, thus we use

.

Thus, we will use the equation


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STUDY GUIDE

Step 3: Substitute with the given number of months and solve for .

Thus, you will be able to save ₱1000 after 3 months.

Example 5: Eduardo receives 5% commission for his sales and additional

allowances of ₱10 000 every month. If he has ₱68 000 sales,

how much would he earn for the month?

Solution:

Step 1: Identify a suitable equation form to be used.

Since the percentage of the commission out of the total sales is constant at

5%, this may be the slope, and since this commission is on top of a fixed

allowance of ₱10 000, the ₱10 000 may be y-intercept. Thus, it is best to use

the slope-intercept form.

Step 2: Set up the equation.

Analyzing the given information, we know that Eduardo’s earnings for the

month depends on the amount of sales he is making, thus his salary for the

month is the dependent variable in this situation, and the amount of sales is

the independent variable.

We let be Eduardo’s salary for the month, and be the amount of the sales.


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STUDY GUIDE

As stated in Step 1, the slope will be the 5% and the y-intercept will

be the ₱10 000.

Thus, we will use the equation

Step 3: Substitute with the given amount of the sales and solve for .

Eduardo’s earning for the month will be ₱13 400.

Try It Yourself!

A car rental company charges ₱4000 a day and then ₱50 for every kilometer that the

car travels. If Luisa rented the car for one day and travelled 15 kilometers, how much

would she pay the company?

1. Write the following equations in slope-intercept form:

a.

b.

c.



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STUDY GUIDE

2. Write the following equations in standard form:

a.

b.

c.

3. Is the equation the slope-intercept form of the equation ?

4. A salesman receives ₱1000 a week for his food and transportation allowance. He

also gets a commission of 15% for all his sales. How much will he receive for the

week if his sales reached ₱22 000?

5. A car park charges ₱30 for the first hour and ₱10 for every succeeding hour. How

much should a car owner pay if his car stayed for 6 hours in the car park?

Do It the Dice Way!

Materials Needed: two dice of different colors, graphing paper, 5 pens of different

colors

Instructions:

1. This activity may be done individually or in pairs.

2. On the graphing paper, make a rectangular

coordinate system.

3. Assign which die will correspond to an x value and

which will correspond to a y value. (In this case, the

red die will correspond to the x value and the blue

die to the y value.)

4. Roll the two dice.

Lesson 5: Graphing Linear Equations

Warm Up!


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STUDY GUIDE

5. Mark the point on the x-axis where the red die landed. Similarly mark the point

on the y-axis where the blue die landed.

6. Connect the two points to make a line.

7. Do steps 3 to 5 four more times with different color of pens.

The activity you just performed is one way of graphing Linear equations.

Recall the following concepts:

The standard form of a linear equation in two variables and , is ,

where , , and are integers, and and are not both equal to 0. By

convention, should also be nonnegative.

The slope-intercept form of a linear equation in two variables and is

, where is the slope and is the y-intercept.

The graph of a linear equation in two variables is a line.

Study the graph of .

Learn about It!


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STUDY GUIDE

What can you say about the two points used to graph the line? How do you think are

these points obtained?

Lines may be graphed in different ways.

Graphing Lines Using the Intercepts

Linear equations can be graphed using their intercepts: x-intercept and y-intercept.

To find the x-intercept, substitute into the linear equation and solve for

To find the y-intercept, substitute into the linear equation and solve for .

The concept applied in Warm Up! is that of graphing a line using its intercepts. The values

that showed up when you rolled the dice were the intercepts you connected to form the line.

Graphing Lines Using the Slope and y-Intercept

Another method of graphing linear equations is by using the slope and the y-intercept.

In this method, you will use the formula:

Definition 5.2: The y-intercept is the point where the line

passes through the y-axis. The coordinates

of the y-intercept follow the form .

Definition 5.1: The x-intercept is the point where the line

passes through the x-axis. The coordinates

of the x-intercept follow the form .


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STUDY GUIDE

Rise is the number of units you will count up from your y-intercept. Run is the number

of units you will count left or right, depending on the sign.

The process starts by plotting the y-intercept along the y-axis. From there, the slope is

used to find the second point by counting a certain number of units up (rise), and

counting a certain number of units to the left or to the right (run).

The slope, as discussed earlier, is a measure of how steep a line is. In addition, the slope is

used to understand the behavior of a given line.

A line is said to be increasing if its slope has a positive value ( ).

A line is decreasing if it has a negative slope ( ).

A line is constant if it has a slope of 0 ( ).

Given the graph of a line, the slope's behavior can be determined without solving for the

exact value of the slope. Starting from the left side of the line, trace the line going to the

right side of the Cartesian plane. If you trace going up, the slope of the line is positive and

it is increasing. If you trace going down, the slope of the line is negative and it is

decreasing.

Example 1: Sketch the graph of the linear equation .

Solution: Note that two points are enough to draw a line. So the goal would then be to

determine two points on the line from the equation. To find points, we

perform the following:

Step 1: Assign a value to one of the variables and substitute it into the linear

equation. Here, let us use .

Let’s Practice!


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STUDY GUIDE

Step 2: Solve for the value of the other variable.

Since corresponds to – , it follows that the point lies on the

line. Be sure to write the numbers in the proper order; should always come

first.

Repeat the same process to find a second point.

Step 3: Assign a value to one of the variables and substitute it into the linear

equation. This time, we let .

Step 4: Solve for the value of the other variable.

Since corresponds to , it follows that the point also lies

on the line.

Now, we can graph the line.


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STUDY GUIDE

Step 5: Plot the two points on the Cartesian plane.

The two points are (–1, 1) and (1, −2).

Step 6: Draw a straight line through the two points.


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STUDY GUIDE

Try It Yourself!

Sketch the graph of the linear equation .

Example 2: Sketch the graph of using its intercepts.

Solution:

Step 1: Find the x-intercept.

Let .

The x-intercept is 8. Its coordinates are (8, 0).

Step 2: Find the y-intercept.

Let .

The y-intercept is −2. Its coordinates are (0, −2).

Step 3: Plot the intercepts and connect them to graph the line.


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STUDY GUIDE

Try It Yourself!

Using the intercepts, sketch the graph of the equation . Compare your

answer to the previous “Try It Yourself!” item.

Example 3: Graph the linear equation using the slope and y-intercept.

Solution:

Step 1: Write the linear equation in slope-intercept form.

Step 2: Find the slope and y-intercept.

or

Step 3: Plot the y-intercept.


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STUDY GUIDE

Step 4: Find the second point using the slope.

The slope, 2, can be written as a fraction.

Step 5: Draw the line by connecting the two points.


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STUDY GUIDE

Try It Yourself!

Using the slope and y-intercept, graph the equation .

Real-World Problems

Example 4: Ronnie walks every Saturday as a form of exercise.

He decided to go farther one particular weekend.

After 15 minutes he was 1 kilometer away from his

house. Maintaining his walking speed, he was 4

kilometers away after an hour. Sketch a graph of

his trip.

Solution: We are given two points in the problem, , i.e., after 15 minutes he was

1 kilometer away, and , i.e., after 1 hour or 60 minutes, he was 4

kilometers away.


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STUDY GUIDE

Plotting the points, and connecting them to draw the graph, we have

Example 5: Using the problem in the previous lesson, let us draw the graph of the given

situation: A car rental company charges ₱4000 a day and then ₱50 for every

kilometer that the car travels. Luisa rented the car for one day and travelled

15 kilometers.

Solution: By analyzing the problem, we can say that it can be written in the slope-

intercept form.

We will let be the total amount of money that Luisa should pay for the rent

and let be the number of kilometers travelled by the car. Also, we will use

50 as the slope since this is the amount charged for every kilometer, and

4000 as the y-intercept since this is the base charge for the car rental.

For this we will use the formula


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STUDY GUIDE

Graphing the equation, we have our y-intercept at 4000. At 15 kilometers,

would be ₱4750. Extending the trip to 25 kilometers, would be ₱5250.

Try It Yourself!

Regina receives ₱400 a day for 8 hours of work as a service crew. She also receives

₱70 for every hour of overtime. Draw a graph showing her salary for a given

number of hours of overtime. (Hint: Use the end of her 8 hours work as the 0 mark

for the x axis.)

1. Sketch the graph of the equation .

2. Graph the line formed by the equation using its intercepts.

3. Graph the linear equation using its slope and y-intercept.

4. A local radio station charges an advertisement for ₱5000 for the first 30 seconds

and ₱1000 for every 15 seconds thereafter. Graph the line of this equation.

5. Mina’s telephone network charges ₱500 a month for 60 minutes of mobile call. It

then charges ₱5 for excess call per minute. Show a graph representing an equation

for the given situation.



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STUDY GUIDE

1. The vertices of a right triangle are , and . Explain why the

point lies on the triangle.

2. Two cyclists are going to climb two hills: Hill A for cyclist 1 and Hill B for cyclist 2.

Points and lie on hill . Hill contains the points and .

Which of the two cyclists would ascend faster?

3. Give five linear equations whose slopes are undefined. Compare the equations.

You are the manager of the school cooperative. Your task is to make an inventory and

replenish the school supplies in the store. For each of the listed type of school supply, list

the cost of each item and the quantity to be purchased. This will be used for future

purchase of the same items. The task should be done in one week time.

School Supply Quantity Cost

ball pen

pencil

intermediate pad

yellow pad

notebook

scissors

glue

ruler

Performance Task

Challenge Yourself!


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STUDY GUIDE

Answer the following situational questions.

1. Suppose that the cost of a ball pen is ₱7.50 and the cost of a pencil is ₱5. If there are

ball pens needed to be replenished and pencils needed to be replenished, write a

linear equation in two variables to model the situation for a ₱1000 budget for ball pens

and pencils.

2. Suppose that there are 50 scissors needed to be replenished at ₱25 each, and that

each glue costs ₱16. If a budget of ₱1890 is given for scissors and glues, how many

glues may be bought with this budget?

3. Suppose that notebooks may be bought in bulk. If you can avail the first 100

notebooks at ₱15 each, and every piece of notebook thereafter costs ₱20, make a

linear equation for this. Sketch the graph of the linear equation.

Performance Task Rubric

Criteria

Below

Expectation

(0–49%)

Needs

Improvement

(50–74%)

Successful

Performance

(75–99%)

Exemplary

Performance

(99+%)

Organization

of data

Below 80% of

the table is

listed and

organized.

80% - 89% of the

table is listed and

organized.

90% - 99% of the

table is listed

and organized.

100% of the

table is listed

and organized.

Formulation

of Equations

There was no

attempt to

formulate

equations in

two variables.

Equations in two

variables were

formulated but

are many

mistakes.

Equations in two

variables were

formulated with

minimal

mistakes.

Equations in two

variables were

formulated and

are accurate.

Timeliness of

Submission

Task was

completed

more than 4

days late.

Task was

completed 3-4

days late.

Task was

completed 1-2

days late.

Task was

completed on

time.

Total


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STUDY GUIDE

Key Terms/Formulas

Key Concept Description

The Rectangular

Coordinate System

The rectangular coordinate plane is composed of two

axes, and .

The point of intersection of the two axes is called the

origin.

As the axes intersect, the plane forms quadrants.

Linear Equations

𝒂𝒙+𝒃𝒚 = 𝒄

Slope

Ways to Graph

- using two points

- using intercepts

- using slope and y-intercept

Forms

Slope intercept Form

y = mx + b

Standard Form

ax + by = c

Wrap-up

2 1

2 1

rise

run

y ym

x x


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STUDY GUIDE

Linear Equations in Two

Variables

A linear equation is an equation with two different

variables, each having an exponent of 1.

The standard form of a linear equation is ,

where a, b and c are real numbers and a and b are not

both equal to 0.

An ordered pair is a solution of a linear equation

if it satisfies the given equation.

Slope of a Line

The slope of a line is the ratio of the change in vertical

distance (rise) to the change in horizontal distance

(run).

The slope can be found using the following

formulas:

.

Writing Linear

Equations in Standard

and Slope-Intercept

Form

The standard form of a linear equation in two

variables and , is ,

where , , and are integers, and

and are not both equal to 0. By

convention, should also be nonnegative.

The slope-intercept form of a linear equation in two

variables and is , where is the slope

and is the y-intercept. This form is especially helpful

when looking for the slope and y-intercept of a linear

equation.

Graphing Linear

Equations

Two points are enough to graph a linear equation.

To find points on a line, assign a value or number to

the first variable, then solving the equation to find

the corresponding value of the second variable.

Graph each point and draw a straight line through

them.

The x-intercept is the point where the line passes

through the x-axis. The coordinates of the x-


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STUDY GUIDE

intercept follows the form (x, 0).

The y-intercept is the point where the line passes

through the y-axis. The coordinates of the y-intercept

follows the form (0, y).

To graph using the slope and y-intercept, plot the y-

intercept first.

In the formula,

, rise is the number of units

counted up from the y-intercept. Run is the number

of units counted left or right from where the rise left

off. The direction is left if the slope is negative, and

right if the slope is positive.

Lesson 1

1.

Key to Let’s Practice!


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STUDY GUIDE

2. ; Quadrant IV

3.

4. 900 meters

Lesson 2

1. and

2. 3

3. c

4.

Lesson 3

1.

2.

3. The line passing through and is steeper. The lines are perpendicular

to each other. The slope of the lines are and

.

4. The slope is 60 which represents the wage per hour.


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STUDY GUIDE

Lesson 4

1.

2.

3. No

4. ₱4750

Lesson 5

1.

2.


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STUDY GUIDE

3.

4.

Baron, Lorraine, et al. Math Makes Sense 8. Canada: Pearson Education, 2008.

Maths Is Fun. “Linear Equations.” Accessed January 6, 2018.

https://www.mathsisfun.com/algebra/linear-equations.html

McGraw-Hill Education. Glencoe Math Volume 1. McGraw-Hill Professional, 2013.

Reference. “What is the Greenwich Meridian?” Accessed February 23, 2018.

https://www.reference.com/science/greenwich-meridian-f657acf27cfa0c42

References

Documents

GRADE 8 | UNIT 3 Linear Equations