Algebra 1 Name · PDF file... Solving Systems of Linear Equations by Graphing - Notes Essential ... Linear Inequalities in Two Variables ... inequality in two variables? Main Ideas

Algebra 1 Name: 5.1 Explain – Solving Systems of Linear Equations by Graphing - Notes

Essential Question: How can you solve a system of linear equations?

Main Ideas/ Questions Notes/Examples

What You Will

Learn

x Check solutions of systems of linear equations. x Determine the numbers of solutions of linear systems. x Solve systems of linear equations by graphing. x Use systems of linear equations to solve real-life problems.

What is a system of

linear equations?

Definition: Create your own system of equations:

What is a solution of a linear system?

Definition: A solution of a system of linear equations is….

1. Prove (2, 5) is a solution for the following system:

2. Is (-2, 0) a solution for the following system?

A system of linear equations can have……. One Solution

The lines intersect.

The solution is __________.

No Solution

The lines are parallel.

Also called an

Infinitely many Solutions

The lines are the same line.

Also called a


3. Use the graph to solve the system of linear equations. Check your solution.

2 52 5x yx y�

� �

Solution: Check:

Solving Systems of

Linear Equations by

Graphing

To Solve by Graphing:

x Graph each equation in the same coordinate plane. 1st: Equations must be in slope-intercept form. 2nd: Graph each line – Begin with the “b”, move with the “m”.

Practice: Solve the system of linear equations by graphing. 4. 3

5y xy x � �

�

5. 12

12

2

4

y x

y x

�

� �

Solution: Solution: 6. 6 6 3

6 6 3x yx y� �

� �

7. 4 3 178 6 34x yx y�

� �

Solution: Solution:


Real-Life Problems

8. A roofing contractor buys 30 bundles of shingles and 4 rolls of roofing paper for $1040. In a second purchase (at the same prices), the contractor buys 8 bundles of shingles for $256. The following system represents the situation for the price per bundle of shingles, , and the price per roll, , of roofing paper.

a. Graph the system.

b. Find the price per bundle of shingles and the price per roll of roofing paper.

Making Connections

9. Which system would be easier to solve by graphing? Explain your choice.

Graphing Using The Calculator

Steps to Solve Systems Using the Calculator: (Round to the hundredths if necessary.) Step 1: Write both equations in slope-intercept form. Step 2: Use “Y1” and “Y2” to enter the equations into. Step 3: Press 2nd, Calc, 5, Enter 3 times. ***Error: No Sign Change Æ the lines are parallel! 10. 0.8 0.9 0

0.5 1x y

x y�

�

11. 4.2 32 0.3

x yx y

�

� �

Solution: Solution:

Algebra 1 Name: 5.2 & 5.3 – Day 1 – Solving Systems of Linear Equations by Substitution and Elimination - Notes

Essential Question: How can you use substitution or elimination to solve a system of linear equations?


What You Will Learn

x Solve systems of linear equations by substitution and by elimination x Use systems of linear equations to solve real-life problems.

Methods to Solve Systems

Method #1

Graphing/Calculator

Method #2

Substitution

Method #3

Elimination

Solving Systems Using Substitution

Steps for Solving Systems Using Substitution:

Step 1: Isolate a variable for the 1st equation Step 2: Substitute and solve into the 2nd equation Step 3: Substitute and solve to find the other variable

1. Equation #1: 𝑦 = −2𝑥 − 9 Equation #2: 6𝑥 − 5𝑦 = −19

What will you be

substituting?

____

will be replaced

with

Substitute and Solve Substitute/Find the Other Variable:

The solution is _______________. Check:

2. Equation #1: 𝑥 = 6𝑦 − 7 Equation #2: 4𝑥 + 𝑦 = −3

What will you be

substituting?

____

will be replaced

with

Substitute and Solve Substitute/Find the Other Variable:

The solution is _______________. Check:

Making Connections: The solutions (ordered pairs) in problems #1 and #2 are where the ……….


Eliminate

What does it mean to “eliminate” something?

Opposite Like Terms

Find the missing numbers that make the equation true. 3. 3 + ____ = 0 4. -2 + ___ = 0 5. -5x + ___ = 0

6. y + ___ = 0 7. 2x + ___ = 0 8. -7y +___ = 0

9. When combining like terms (adding/subtracting), what must the terms look like to equal “0”?

Solving Systems Using Elimination

Steps for Solving Systems Using

Elimination:

Step 1: “Line up” like terms. (Standard Form works best. 𝐴𝑥 + 𝐵𝑦 = 𝐶) Step 2: Multiply, if necessary, one or both equations to get opposite coefficients for like terms. Step 3: Add or Subtract the equations together and solve. Step 4: Substitute and solve to find the other variable.

Practice: Solve the following systems using elimination.

10. 𝑥 + 3𝑦 = 17 −𝑥 + 2𝑦 = 8 Add/Subtract the equations & solve: Substitute & solve to find the other variable:

Solution:

11. 2𝑥 + 3𝑦 = 10 Solution: −2𝑥 − 𝑦 = −2 Add/Subtract equations & solve: Substitute & solve:

12. −𝑥 + 𝑦 = 3 Solution: −3𝑥 − 𝑦 = 1 Add/Subtract equations & solve: Substitute & solve:

Summary: When solving systems of equations by……. SUBSTITUTION,

look for…….. ELIMINATION,

look for…..


Essential Question: How can you use substitution or elimination to solve a system of linear equations?


Methods to Solve Systems of Equations

Graphing/Calculator Substitution Elimination

Special Cases When all the variables cancel completely out…. Getting a FALSE statement is NO SOLUTION (a TRUE statement is INFINITE SOLN)

Solving Systems Using Substitution or Elimination Solve each system of equations ~ Choose the best method

1. 𝑥 = 3 − 𝑦 Substitution/Elimination 5𝑥 + 3𝑦 = −1 Solution:

2. 𝑥 − 𝑦 = 6 Substitution/Elimination 𝑥 + 𝑦 = 8 Solution:

Making Adjustments: Sometimes, you may need to rearrange the variables in an equation or multiply each term in an equation by a number in order to make one of the methods work.

Substitution ~ Isolate a Variable Look for Æa term with a coefficient of ______

3. 𝑦 − 1 = 4𝑥 Solution: 3𝑦 = 3𝑥 − 6

Elimination ~ Multiply

4. 4𝑥 + 2𝑦 = 2 Solution: −2𝑥 + 𝑦 = 5

Algebra 1 Name: 5.2 & 5.3 – Day 2 – Solving Systems of Linear Equations by Substitution and Elimination - Notes Practice: Solve each system of equations by choosing the best method. 5. 2𝑥 − 3𝑦 = 16 Substitution/Elimination? 2𝑥 + 4𝑦 = 2 Solution:

6. 𝑥 + 3𝑦 = 5 Substitution/Elimination? 5𝑦 + 3 = 4𝑥

Solution:

7. −3𝑥 + 3𝑦 = 4 Substitution/Elimination? 𝑦 − 3 = 𝑥 Solution: ____________

8. 𝑥 + 4𝑦 = 3 Substitution/Elimination? −𝑥 − 4𝑦 = −3 Solution: ____________

Application: 9. An adult ticket, 𝑥, to a museum costs $3 more than a children’s ticket, 𝑦. When 200 adult tickets and 100 children’s tickets are sold, the total revenue is $2100. What is the cost of a child’s ticket?

Let 𝑥 = Solve: Interpret the Solution:

Let 𝑦 =

System of Equations: Substitution/Elimination? 10. The graph represents the average salaries of classroom teachers in two school districts. Solve the system using the method of your choice.

a. What year were the average salaries in the two districts equal?

b. What was the average salary in both districts in that year?

which means the lines are…..

which means the lines are…..

Algebra 1 Name: 5.6 Explain – Linear Inequalities in Two Variables - Notes

Essential Question: How can you write and graph a linear inequality in two variables?


What You Will

Learn

x Check solutions of linear inequalities. x Graph linear inequalities in two variables. x Write linear inequalities in two variables. x Use linear inequalities to solve real-life problems.

What is a linear

inequality?

Definition: An inequality uses the following symbols:

� - __________________ ! - ____________________

d - __________________ t - ____________________

What is a solution of

a linear Inequality?

Definition: Practice: Tell whether the ordered pair is a solution of the inequality.

1. � �5; 3, 2x y� ! 2. � �2; 5, 3x y� t 3. � �2 5; 2, 3x y� � t � �

Graphing Linear Inequalities Steps to Graph an Inequality: Step 1: Put the inequality in slope-intercept form. Step 2: Graph the inequality using the slope and y-intercept. Step 3: Graph the boundary line. Step 4: Shade to represent all of the solutions. Practice: Graph the inequality in a coordinate plane.

4. 4y � 5. 1x d � 6. 3 1y x� � m = Line: DASHED/SOLID

b = Shade: ABOVE/BELOW Can solutions be on the line? Can solutions be on the line? Can solutions be on the line?

Is (0, 0) a solution? Is (-1, 0) a solution? Is (-1, 0) a solution?

Algebra 1 Name: 5.6 Explain – Linear Inequalities in Two Variables - Notes 7. 1y xt � � 8. 9. 2x y� � m = Line: DASHED/SOLID

b = Shade: ABOVE/BELOW

Writing an

Inequality

10. Write an inequality that represents the graph. m = b = Line: Shading: Inequality:

11. An ice cream truck can carry at most 75 gallons of ice cream. The table shows the inventory on the truck. Write an inequality that represents the numbers of gallons of strawberry and banana ice cream on the truck.

Real-Life

12. An online store sells digital cameras and cell phones. The store makes a $100 profit

on the sale of each digital camera x and a $50 profit on the sale of each cell phone y. The store wants to make a profit of at least $300 from its sales of digital cameras and cell phones.

a. Write an inequality that represents how many digital cameras and cell phones they must sell.

b. Graph the inequality.

b. Identify and interpret two solutions of the inequality.

Algebra 1 Name: 5.7 Explain – Systems of Linear Inequalities - Notes

Essential Question: How can you graph a system of linear inequalities?


What You Will Learn

x Check solutions of systems of linear inequalities. x Graph systems of linear inequalities. x Write systems of linear inequalities. x Use systems of linear inequalities to solve real-life problems.

What is a system

of inequalities?

Definition: Remember:

x When graphing, inequalities may be graphed from slope-intercept form or standard form.

x When solving an inequality….You MUST “flip” the inequality sign, when multiplying or dividing by a negative.

x When graphing an inequality…….

What is a solution to a

system of inequalities?

Definition: An ordered pair that makes ALL inequalities true. (Where the shaded regions overlap.) Practice: Tell whether the ordered pair is a solution of the system of linear inequalities. 1. (2, 3); 4

2 4y xy x

t �

d �

2. (0, 4); 45 3

y xy xd � �

t �

Graphing Systems of Inequalities

Step 1: Graph each inequality in the same coordinate plane. Step 2: Finding the solution…..The region where the shading overlaps or intersection, of the graphs of the inequalities.

Practice: Each system of inequalities has been graphed except for the shading. Shade the appropriate region for each inequality and then identify the region that the solutions are in. 3. t 4. xy �! Region: Region:

Algebra 1 Name: 5.7 Explain – Systems of Linear Inequalities - Notes Practice: Determine if the given ordered pair is a solution to the inequality. 5. (4, 1) YES NO 6. (0, 5) YES NO

Writing: Explain how you know if an ordered pair is a solution to a system of linear inequalities given a graph. Practice: Graph the system of linear inequalities.

7. 2

3yy x! �

d

8. 22

y xy x

t �

� � �

9. 2 3 61 2

x yy x

� �

� t �

10. Write a system of linear inequalities represented by the graph.

11. Real-Life: You have at most 8 hours to spend at the mall and at the beach. You want to spend at least 2 hours at the mall and more than 4 hours at the beach.

a. Write a system that represents the situation. Let be the number of hours at the mall and let be the number of hours at the beach.

b. Graph the system.

c. How much time can you spend at each location?

Algebra 1 Name: 6.1 Explain – Properties of Exponents Day 1 - Notes

Essential Question: How can you write general rules involving properties of exponents?


What You Will Learn

x Use zero and negative exponents. x Use the product of powers and quotient of powers properties of exponents. x Solve real-life problems involving exponents.

Exponent

Review

x A power is a product of repeated factors. x The base of a power is the common factor. x The exponent of a power indicates the

number of times the base is used as a factor.

Zero Exponents Negative Exponents

Use your calculator to evaluate the following: 1. 2. 3. Write a conjecture based on the previous three problems:

=

***Simplified polynomials may NOT have negative exponents.***

Using Zero and Negative Exponents

Practice: Evaluate the expression.

1. 03 2. � �02� 3. 43� 4. � � 34 ��

5. 3

025

�

6. 2

33

2

�

�

� 7. 1

047

�

� 8.

� �

1

035

�

�

Practice: Simplify the expression. Write your answer using only positive exponents. 9. 0z 10.

8a� 11. 0 26a b�

12. 4 014m n� 13. 2 3

03 rs

� �

14. 3 3

1 5 02

8ab c

�

� �


Product of Powers Property

To multiply powers with the same base….

x Multiply coefficients, ADD exponents.

What it looks like: 3x3(4x6)

What it means:

What it is:

Practice: Simplify the expression. Write your answer using only positive exponents. 15. x4 y x7 16. (m3)(m3) 17. r2 • r 18. 19. x • x7 • x 20. (ab4)(ab2c) 21. (4x2y3)(6x12y) 22. am · an

Quotient of Powers Property

To divide powers with the same base….

x Divide coefficients, SUBTRACT exponents.

What it looks like:

What it means:

What it is:

Practice: Simplify the expression. Write your answer using only positive exponents.

23. 2

7

xx

24. 5

3

mm

25. 5

3

96xx

26. yxyx

4

5

318

27. 4

2

420xx�

28.

29.

30. A rectangular prism has length x, width ,2x and height .

3x Which of the expressions represent the volume of the

prism? Select all that apply. A. 1 36 x� B. 1 36 x� � C.

D. 1 1 32 3 x� �x x


Essential Question: How can you write general rules involving properties of exponents?


What You Will Learn

x Use the properties of exponents. x Solve real-life problems involving exponents.

STAAR Algebra 1 Reference Materials

Power of a Power Property

rule: To raise a power to a power….

x Multiply exponent to exponent. What it looks like:

What it means:

What it is:

Practice: Simplify the expression. Write your answer using only positive exponents. 1. � �35q 2. � �24a� 3. 4.

Power of a Product and a Quotient Property

Practice: Simplify the expression. Write your answer using only positive exponents.

4. � �44d� 5. � � 33 f �� 6. 34

x

�§ ·¨ ¸© ¹

7. (

) 8. (

)


App

licat

ion

9. Find the area of a square with a side length of units.

10. Find the area of the circle. Leave in terms of .

11. Which expression is equivalent to the volume of the cylinder shown, where is the radius and is the height?

Documents

Algebra 1 Name · PDF file... Solving Systems of Linear Equations by Graphing - Notes Essential ... Linear Inequalities in Two Variables ... inequality in two variables? Main Ideas