Chapter 3: Lines and Linear Equations€¦ · Lesson 1: Graphing Lines Today, we will be learning how to graph lines in slope-intercept form. Remember, in Chapter 2 you learned slope-intercept

5/6/2019 Workbook Revised Chapter Review - Google Docs

https://docs.google.com/document/d/1jtAm5lh2hSeWqrP21f70RlqYmKX7gNwlzfqwYzuGOhY/edit 61/335

Chapter 3: Lines and Linear Equations

Lesson 1 Graphing Lines Page 62

Lesson 2 Writing Linear Equations Page 66

from Graphs

Lesson 3 Horizontal and Vertical Lines Page 70 Lesson 4 Creating & Solving Linear Page 74

Inequalities

Lesson 5 Graphing Linear Inequalities Page 78

Lesson 6 Modeling Linear Inequalities Page 82

Lesson 7 Chapter Review Questions Page 86

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Lesson 1: Graphing Lines

Today, we will be learning how to graph lines in slope-intercept form. Remember, in Chapter 2 you learned slope-intercept form is a linear equation that is in the form of y=mx+b , where m is the slope, and b is the y-intercept.

Graphing Lines in Slope-Intercept Form Step 1. Identify the y-intercept (the b value).

This is the value on the y-axis where you will b egin your graph. Step 2. Plot the y-intercept on the graph. Step 3. Identify the slope of the line (the m value).

This is the ratio of the change in y to the change in x, or how much to “rise” versus how much to “run”.

Step 4. Use the slope to rise and run from the y-intercept and m ove to the next point on the line.

Step 5. Connect the points using a straightedge.

Exercise #1: Consider the linear equation x y = 32 + 4

Identify the y-intercept. The y-intercept is b = ____ or ( , ). Plot the y-intercept. Identify the slope. The slope is m = _____. This means we move _________ 2 units and ________ 3 units from the y-intercept. Plot the next point on the line, then connect the points with a straight edge. Think: What if the slope was ? Would anything change when graphing the line?− 3

2

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Exercise #2: Graph the following linear equations on the set of axes provided. a) xy = 2

1 − 3 b) x y = − 2 + 1

Determining an appropriate scale is very important when it comes to graphing linear equations. Take a look at the equation in Exercise #3 and determine the most appropriate scale to graph the line. Exercise #3: Graph the line of the equation 0x 5 y = 5 + 2

a) What did you choose as the scale? Why? b) Did you scale both axes or just one? Why?

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Now, we will look at some word problems where you will set up an equation and graph the line. In some cases, you may need to place the equation in slope-intercept ( y=mx+b ) form before you graph.

Exercise #4: Josh walks 5 blocks from school to his friend’s house. Then, he walks home from his friend’s house at a constant rate of 1 block per minute.

a) Write an equation to represent the total number of blocks, y, Josh will walk x minutes after leaving his friend’s house. b) Graph the equation. Determine an appropriate scale and set of axes for the graph.

Think: What units are represented on the y-axis? What units are represented on the x-axis? What units are represented by the slope? Exercise #5: Valerie is saving money for a new bicycle. She has already saved $150 and is able to save $50 each week towards the bicycle. Create an equation and graph the line that represents this situation.

a) Based on your graph, how much will Valerie have saved after 3 weeks? b) If the bicycle costs $450, how many weeks will it take Valerie before she is able to buy the bicycle?

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Lesson 1 Extra Practice

EP1 . Graph the line of the equation EP2 . Graph the line of the equation 3y + 2x = 15 6y - 3x = 12

EP3 . Brittany is running a marathon. She takes her pulse while running and notices that there are 15 beats over a ten second interval.

a) What is Brittany’s pulse, in beats per minute ? b) Sketch a graph of Brittany’s pulse where x represents the number of minutes, and y represents the number of beats. c) What is the equation of the line you sketched?

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Lesson 2: Writing Linear Equations from Graphs

In the previous lesson, you learned how to graph lines given an equation in slope-intercept form. In this lesson, you will be determining linear equations given the graph of the line. You will be able to use the point-slope formula, as well as slope-intercept form to help you create equations based on graphs.

The two methods we will use to determine linear equations from graphs are:

Point-Slope Formula: Slope-Intercept Form: (x ) y − y1 = m − x1 y x = m + b

To use point-slope formula, you must know: To use slope-intercept form, you must know:

- the slope of the line - the slope of the line - any point on the line - the y-intercept of the line

Exercise #1: Given the graph below, determine which method you would use to find the equation of the line. a) Find the equation using the b) Find the equation using the point-slope method: slope-intercept method:

c) What do you notice about your answers using either method?

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Exercise #2: Use either method to find the linear equations for each of the following graphs.

a) b)

c) d)

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EP1 . Determine the equation of the line based on the given information.

a) If the x-axis scale is 1, b) If the x-axis scale is 2, and the y-axis scale is 1 and the y-axis scale is 5 c) If the y-axis scale is 0.5, d) If the y-axis scale is 10, and the x-axis scale is 1 and the x-axis scale is .5

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EP2 . Maura thinks the equation of the line below is 2y - 6x = 12. Explain whether Maura is correct or incorrect.

EP3 . Based on the line graph below, find the exact coordinate of the y-intercept.

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Lesson 3: Horizontal & Vertical Lines

Most linear equations satisfy the form y = mx + b . However, some equations exist where that may not be the case. Today, we will take a look at these special circumstances and learn how to write equations of horizontal and vertical lines.

Exercise #1: On the grid below, plot the points (3, 5) and (-2, 5), then graph the line that passes through these two points.

a) State two more points on the graphed line. b) Do all of these points have something in common? c) How could you write the equation of this line?

Exercise #2: On the grid below, plot the points (-3, 2) and (-3, 7), then graph the line that passes through these two points.

a) State two more points on the graphed line. b) Do all of these points have something in common? c) How could you write the equation of this line?

Horizontal Lines Vertical Lines For any horizontal line, the equation is: For any vertical line, the equation is:

y = b, x = a, where b is the y-intercept. where a is the x-intercept.

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Exercise #3: For each of the following, write the equation of the line.

a) b) c)

Equation: ________________ Equation: ________________ Equation: ________________ d) e) f)

Equation: ________________ Equation: ________________ Equation: ________________ Exercise #4: What is the equation of a Exercise #5: What is the equation of horizontal line passing through the a vertical line passing through the point point (2, 6)? (-1, 4)? Exercise #6: What is the equation of a Exercise #7: What is the equation of a line line parallel to the x-axis and passing parallel to the y-axis and passing through through the point (6, 12)? the point (-3, 0)?

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EP1. For each of the following, write the equation of the line. a) b)

Equation: ______________________ Equation: ______________________

c) d)

Equation: ______________________ Equation: ______________________

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EP2 . Write the equation of a line...

a) ...parallel to the x-axis and passes through the point (5, -2).

b) ...perpendicular to the x-axis and passes through the point (-13, 20).

c) ...that has a slope of zero and passes through the point (5, 14).

d) ...parallel to the y-axis and passes through the point (2, -8).

e) ...that has an undefined slope and passes through the point (-2, 4).

f) ...that passes through the points (6, 8) and (9, 8).

g) ...that passes through the points (-4, 2) and (-4, 15)

h) ...perpendicular to the x-axis and passes through the origin.

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Lesson 4: Creating & Solving Linear Inequalities

In this lesson, we will practice setting up inequalities from real-life scenarios. This will lead us into graphically representing these inequalities in the next lesson. Exercise #1: A construction worker earns $53 per hour. She buys lunch o� of the lunch-truck for $12 each day. How many hours must she work in order to earn more than $360 each day, after paying for lunch? Exercise #2: A taxi service charges $6.00 as a flat fee plus an additional $1.00 per mile driven. How many miles can a customer with $20 a�ord to travel? Exercise #3: Admission to the carnival is $5.00 plus $1.25 for each ride. How many rides can a child go on if he arrives at the carnival with $12.00?

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In Exercises #1 - 3, the inequalities were all one variable inequalities. We will now take a look at some two variable inequalities. Exercise #4: Isolate y for each of the following. a) x y 6 4 + 2 ≥ 1 b) x y 2 2 − 3 ≤ 1 c) x y 4 6 + 6 > 2 d) y x 6 − 8 − 4 < 1

Exercise #5: What is the largest integer a , that satisfies the following inequality?

a a 0 3 − 5 + 6 > 1

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EP1 . Fill in the table below with the correct inequality ( ) that is used with the, , ,< > ≤ ≥ corresponding words or phrases.

Word/Phrase Inequality

a maximum of more than

at least a minimum of

below no more than

at most above

fewer than EP2 . For each of the following scenarios set up and solve an inequality that represents the given situation. a) Grace’s hair is currently 5 inches long, and grows at a rate of 1.5 inches per month. Grace wants her hair to be at least 14 inches long. What is the least number of months she must wait? b) Three times a number x, subtracted from 18 is less than -90. What is the smallest possible value of x?

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EP3 . Isolate y for each of the following, placing the inequality in slope-intercept form. a) 2x 8y 0 − 1 + 1 > 9 b) 8x 5y 68 2 + 3 ≥ 1 c) x y 1

10 + 21 < 2 d) .45x .8y 7 0 − 1 > 2

e) x y x 5 + 2 > 3 f) x 2y (2x ) 0 8 − 1 > 4 + 5 − 2

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Lesson 5: Graphing Linear Inequalities

At this point, you have become an expert at graphing linear equations on a coordinate plane. Now, we will graph linear inequalities on the coordinate plane as well.

Exercise #1: Consider the inequality x y < 21 + 3

a) Do the following points satisfy the inequality? (-4, 8) (2, 4) (3, -1)

b) Graph the inequality on the grid. Begin by graphing a dashed or dotted line of the equation . Why is the line dashed?x y = 2

1 + 3 c) Since y “is less than” (<), we will shade the area below the line (downward on the y-axis). Also, notice how part a makes sense with this graph.

Exercise #2 : Graph the inequality x y > 2 − 3 Exercise #3: Graph the inequality − x y ≥ 21 + 4

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Exercise #4: Which of the following represents the graph of the inequality . If thexy ≥ 2

1 graph is not accurately represented, state why it is not correct.

Describe any error on this graph. Describe any error on this graph.

Describe any error on this graph. Describe any error on this graph.

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EP1 . Graph the inequality x y > − 32 + 3 EP2 . Graph the inequality y ≤ 5

EP3 . Graph the inequality x > − 3 EP4 . Graph the inequality x y < 5

1 + 4

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EP5 . Graph the inequality y x 2 3 + 4 > 1

EP6 . Graph the inequality x y x 7 − 2 ≥ 4 + 6

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Lesson 6: Modeling Linear Inequalities

Graphs are often used in the real world to illustrate data and statistics in a user friendly way. Graphs of linear inequalities are useful because they show all of the possibilities within a given scenario. It will be up to you to determine appropriate solutions based on the context of the problem. Exercise #1: A farmer in upstate New York has cattle and horses. Each cow eats 20 pounds of food per week, and each horse eats 30 pounds of food per week. If the farmer can a�ord to buy 210 pounds of food each week, set up and graph an inequality representing all of the possible combinations of cattle, x, and horses, y, the farmer can own.

Exercise #2: It costs $1.75 to buy a pack of gum, and $5.25 to buy an energy drink. If you have $21 to spend, set up an inequality and graph all the possible combinations of gum, x, and energy drinks, y, you could buy.

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Exercise #3: At a fruit store, watermelons weigh 8 pounds and cantaloupes weigh 5 pounds. The weight of the bin that holds the watermelons and cantaloupes must be less than 90 pounds. Set up an inequality and graph all the possible combinations of the amount of watermelons, x, and cantaloupes,y, the bin can hold.

Exercise #4: At the surf shop, boardshorts cost $30 and t-shirts cost $20. If the surf shop wants to sell at least $400 worth of boardshorts, x, and t-shirts, y, set up and graph an inequality that shows the possible combinations that would allow the shop to reach their goal.

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EP1 . During a sale, all video games cost $25 and all movies cost $10. If Zoe wants to buy video games and movies with some or all of her $200 in birthday money, set up and graph a linear inequality to represent all of the possible combinations Zoe can buy.

a) Is the point (-1, 1) in the solution set? Explain your answer. b) Based on your answer above, what would be a reasonable domain for this problem? Explain your reasoning. c) List three possible combinations in the solution set.

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EP2 . The manager of an electronics store projects that the revenue from the sale of televisions and cameras should be at least $18,000 per day. a) If televisions sell for $550 each and cameras sell for $300 each, set up an inequality to represent how many televisions, t, and cameras, c, must be sold in order to make the desired amount. b) The store manager sells his entire stock of 30 televisions. How many cameras must be sold to reach the desired revenue? EP3 . Sean incorrectly graphed the inequality on the graph below.y − x − 2 < 8

a) Explain Sean’s mistake while graphing this inequality.

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Chapter Review

Part I Questions : For each statement or question, choose the word or expression that, of those given, best completes the statement or answers the question. CR1. What inequality is represented by the graph below?

1) x y < 2 + 1

2) x y < − 2 + 1

3) x y < 21 + 1

4) x y < − 2

1 + 1 CR2 . What is an equation of the line that passes through the point (-5, 6) and has a slope of -2?

1) x 6 y = − 2 − 1 3) x 4 y = − 2 − 1

2) x y = − 2 − 4 4) x 6 y = − 2 + 1 CR3 . If point (-3, -7) is on the line whose equation is , what is the value of ?x y = 5 + b b

1) 2 3) 6 2) 4 4) 8

CR4 . Which equation represents the line that passes through the points (-2, -5) and (2, 7)?

1) x y = 3 + 1 3) x y = 4 + 2

2) xy = 3 − 1 4) x y = 4 − 2

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Open Response Questions : For each question, clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. CR5 . On the set of axes below, draw the graph of the equation x y = − 3

2 + 4

Is the point (2, 3) a solution to the equation? Explain your answer based on the graph above.

CR6. Dulce and Nexer were doing their algebra homework. They were asked to write the equation of the line that passes through the points (−6, 7) and (9, 2). Dulce wrote

and Nexer wrote . Justify why both students are correct. (x ) y − 7 = − 31 + 6 x y = − 3

1 + 5 CR7 . Liam wants to buy cookies and cupcakes for a birthday party. Cookies cost $1.90 each and cupcakes cost $2.60 each. If he only has $40 to spend on cookies and cupcakes, write an inequality that represents the situation where he buys x cookies and y cupcakes.

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CR8 . Graph the inequality on the set of axes below.(x 0) y − 3 < − 2

1 + 1

CR9 . Catherine has two jobs. She earns $10 per hour walking dogs in the neighborhood and $13 per hour working at the public library. Write an inequality to represent the number of hours, x, walking dogs and the number of hours, y, working at the public library that Catherine will need to work to earn a minimum of $250. Catherine walked dogs for 5 hours. Use the inequality to find the number of full hours she must work at the public library to reach her goal of $250.

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CR10 . Jeremiah and his friends are at a minor league baseball game. Sodas cost $1.25 each and cheeseburgers cost $5 each. If Jeremiah and his friends can spend at most $15, set up an inequality to represent the scenario. Graph the inequality on the grid below. Be sure to label your axes.

How many purchasing combinations are possible if Jeremiah and his friends buy at least one soda and at least one cheeseburger?

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Documents

Chapter 3: Lines and Linear Equations€¦ · Lesson 1: Graphing Lines Today, we will be learning how to graph lines in slope-intercept form. Remember, in Chapter 2 you learned slope-intercept