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Graphing Linear Equations Review Graph each equation.

1. y = 3x – 1 2. y = 1

32x

3. 1236 yx 4. y x 2

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What do we know about parallel and perpendicular lines?

Parallel lines –

Perpendicular lines –

What do we know about the sides of a rectangle?

What do we know about the angles of a rectangle?

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Question A: Coordinates of points B, C, E, F, and G

B: C: E: F: G:

Question B: Find the slopes of each segment for both rectangles

Rectangle ABEF Rectangle BCFG

Segment Slope Segment Slope

Question C: Which segments are parallel? How do you know?

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Question D: Which segments are perpendicular? How do you know?

Question E

Question E: Slopes of Line Segments

Rectangle ABCD Parallelogram EFGH Parallelogram PQRS Rectangle TUVW

AB EF PQ TU

BC FG QR UV

CD GH RS VW

AD EH PS TW

Determine whether the following lines are parallel, perpendicular, or neither. Find the slopes of the points given to

find your answer.

1. Line 1: (3,2) and (-1, 6) 2. Line 1: (3,-4) and (0,-6)

Line 2: (5,10) and (8,13) Line 2: (6,10) and (15,16)

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Decision Making A new amusement park just opened in Rochester called “Marsh Madness.” The park offers its customers two admission options to choose from: 1. For each option, state the y-intercept and slope, and write an equation to describe the situation where y is the total cost and x is the number of rides the customer goes on.

Option 1 Option 2

y-intercept

slope

equation

2. Graph both equations on the grid below:

Option 1 Customers pay $6 to enter the park and $2 per ride they go on.

Option 2 Customers pay $2 to enter the park and $3 per ride they go on.

Co

st

# of rides

5

3. How much would it cost a customer for each option given:

# of Rides Option 1 Option 2

0 rides

3 rides

4 rides

5 rides

4. You have $40 to spend. Which option would allow you to ride the most rides? Show your work and explain your reasoning. 5. For what number of rides would the two options cost the same? Explain. 6. Based on your findings in the graph and table, when would it be more economical for you to choose option 1 over option 2? Explain.

Solving Systems of Equations by Graphing The equation 2x + 1 = 9 has a solution of x = 4. This is the solution of the equation because when 4 is substituted for x the equation makes a true statement.

A line is formed by an infinite number of ordered pairs that make the equation true. Graph the two equations on the left on the graph provided.

What is unique about the point (4, -3)?

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Graph the next two equations on the graph provided. These lines are _______________________.

What do you look for in order to identify this type of lines? Since the lines never ___________________ this system of linear equations has __________________.

Graph and find the solution: 2x + y = 6 y = -2x + 6

What is the solution to this system? Explain why you came to this conclusion. If the lines have the same slope and y-intercept the solution is_____________________________________

Summary: Intersecting Lines Parallel Lines Coincident (Same) Lines

Number of Solutions

Slopes

Y-Intercepts

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1. Solve by graphing. 2. Solve by graphing y = 3x – 4 -5x + y = -2 y = -3x + 2 -3X + 6Y = -12

3. Solve by graphing. 4. Solve by graphing. -3x + 3y = 4 4x – 3y = -9 -x + y = 3 2x + 3y = 9 5. Solve by graphing. 6. Solve by Graphing. 2x + 2y = 10 y = 5x – 7 y = -x + 5 -3x – 2y = -12

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By graphing, determine if the following system has one solution, no solution, or infinitley many solutions. 7. and 8. and 9. and

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Solving Systems using Substitution. So far we have discussed graphing as a method to solve systems of equations. However, graphing is not the only way to solve a system. There are 2 algebraic methods we will study. The first of those methods is called substitution. Example 1

62

2

yx

x

Solution: ( ____ , ____ ) Example 2

1

843

y

yx

Solution: ( ____ , ____ ) Example 3

12

20

yx

x

Solution: ( ____ , ____ )

Example 4

943

1

yx

xy

Solution: ( ____ , ____ )

Example 5

6

1023

yx

yx

Solution: ( ____ , ____ ) Example 6

122

53

xy

xy

Solution: ( ____ , ____ ) Example 7

62

52

1

yx

xy

Solution: ( ____ , ____ ) Example 8

2433

8

yx

yx Solution: ( ____ , ____ )

11

5

2 2 14

x y

x y

3 6 5

2 4 6

x y

y x

Guided Practice Solve the System by: Graphing Substitution

1. 152

5

yx

xy

2.

3.

When solving using the Substitution Method which variable would you solve for? Explain why you chose that variable.

4. 5. 6.

843

122

yx

yx

753

8102

yx

yx

12

205

yx

yx

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Solve the following systems of equation using the Substitution Method. 7. 8. 9.

653

4

xy

yx

)2(21

52

xy

xy

1552

92

yx

yx

10. Find EVERY mistake this student made when solving this problem using substitution. Then solve the

system correctly.

The Problem: 243

842

yx

yx Mistakes Solve the System

Step One:

243

42

yx

yx

Step Two: 24)42(3 yy

Step Three: 24126 yy

Step Four: 2122 y

Step Five: 102 y

Step Six: 5y

Step Seven: 8)5(42 x

Step Eight: 14x Solution: )14,5(

Solve each system using substitution. Write your answer as an ordered pair.

1. 127

33

yx

yx 2.

72

12

yx

yx

3. 155

03

y

yx 4.

1

42

yx

yx

13

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Solving Systems by Elimination (Combination) Sometimes neither equation can easily be solved for one variable, but you are still able to solve the system. Elimination Method: ________________________________________________________________ Type I: Add to eliminate The equations are written in standard form (Ax + By = C), this allows the x and y variables

to be aligned in the two equations. Notice the coefficients of x have the same absolute value (-4 and 4). What happens when you add -4 and 4? What happens if you add –4x and 4x? If you add the two equations together the result is 6y = -36 Solve for y, y = -6 Substitute -6 for y in one of the original equations and solve for x, 4x + 8(-6) = -24, x = 6 Write your final answer as a point (6, -6)

Type II: Multiply one equation Notice the coefficients of neither x or y has the same absolute value. Equations are balanced and maintain properties of equality. Is there anything we can do to one of the coefficients of one equation to make it the same

as the coefficient of the other equation? Multiply the first equation by 2, both sides, to give you 10x + 2y = 18 If you subtract the two equations together, to eliminate the x’s, the result is 9y = 36 Solve for y, y = 4 Substitute 4 for y in one of the original equations and solve for y, 5x + 4 = 9, x = 1 Write your final answer as a point (1, 4)

Type III: Multiply both equations Notice the coefficients of neither x or y has the same absolute value. Equations are balanced and maintain properties of equality. Choose one variable and determine the Least Common Multiple (LCM) The LCM of the x’s is 12 and the LCM of the y’s is 15 To eliminate the x’s,

o Multiply the first equation by 3, both sides, to give you 12x – 9y = 33 o Multiply the second equation by 4, both sides, to give you 12x – 20y = -44

If you subtract the two equations together, to eliminate the x’s, the result is 11y = 77 Solve for y, y = 7 Substitute 7 for y in one of the original equations and solve for y, 4x – 3(7) = 11, x = 8 Write your final answer as a point (8, 7)

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16

17

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Word Problems Process Notes

VISUALIZE – Draw a picture of what is happening in the problem.

THINK – What is the question? What are you being asked to do? Rewrite the question.

EXPLAIN – What is your thought process? How will you solve the problem?

First I…

Then I…

Finally I…

SOLVE – What is your mathematical plan? Show your work.

Step 1

Step 2

Step 3

WRITE a conclusion. Be sure your conclusion is reasonable and answers the original question.

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Notes- Steps to Solving Systems Applications Define Variables Set up table/write equations Choose method and Solve Check answers

Practice – Solving Systems Applications

1. The sum of two numbers is 70 and their difference is 24. Find the numbers.

2. Find two numbers whose sum is 18 and whose difference is 22.

3. Three times one number equals twice a second number. Twice the first number is three more than the second number. Find the numbers.

4. The length of Sally’s garden is 4 meters greater than three times the width. The perimeter of her garden is 72 meters. What are the dimensions of the garden?

5. Two angles are supplementary. The measure of one angle is 10 more than three times the other. Find the measure of each angle.

6. Nancy’s Nut shop sells cashew for $3.10 per pound and peanuts for $1.95 per pound. How many pounds of peanuts must be added to 15 pounds of cashews to make a mix that sells for $2.70 per pound?

7. Carmen sold play tickets. Each adult ticket cost $4.00 and each student ticket cost $2.00. The total number of tickets sold was 260, and the total income was $700. How many of each kind of ticket was sold?

8. Josh is doing a chemistry experiment that calls for a 30% solution of copper sulfate. He had 40 mL of a 25% solution. How many milliliters of 60% solution should he add to obtain the required 30% solution?

9. Anna has 4 times as many nickels as dimes in her bank. Her bank account contains $7.20. How many

nickels and dimes are in Anna’s bank?

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21

22

23

x

y

x

y

Warm-up Graph the following lines on the coordinate plane.

1. 2.

Graph the following inequalities on the number line. 3. x > 4 4. 5. 8 > x

From above, you can see the graph of a linear equation in two variables is a line. The graph of a linear inequality (one variable) is a ___________ ______________. On a number line > and < use an open dot. On a coordinate plane > and < use a _____________ _________________. On a number line and use an closed dot. On a coordinate plane and use a _____________ _________________. Ex. #1 Graph x = 3 Ex. #2 Graph x < 3 A solution is any A solution is any point point on the line. in the shaded region. The solution does not

include points on the line.

Ex. #3 Graph Ex. #4 Graph

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When y is by itself: < shades down and > shades up. Ex. #5 Graph Ex. #6 Graph Are the points (3, 2), (-2,-2) , (-1, -3) solutions to example 6? How do you know graphically? How do you know algebraically? In the problem about the eighth-graders selling t-shirts for $5 and caps for $10, they had to buy those t-shirts for $2 and caps for S5. If they only had $200 to buy items to sell, how many of each could they afford? Write an inequality to represent this situation. _______________________________________ If you only bought shirts, how many could you buy? If you only bought caps, how many could you buy? Express these numbers as ordered pairs. These are your x and y intercepts. Use them to graph your inequality. Test a point to decide which side to shade. List possible solutions to your inequality. Why are we only looking at the first quadrant?

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You try: Suppose you plan to spend no more than $24 on meat for a cookout. At your local market, hamburger costs $3.00/lb and chicken wings cost $2.40/lb. Find three possible combinations of hamburger and chicken wings you can buy Inequality: Intercepts: Graph Possible Combinations: Steps: Graph each line

Solid or dotted line Shade above or below or : solid line or : above (only if y is positive) or : dotted line or : below (only if y is positive)

THE SOLUTIONS ARE POINTS IN THE AREA THAT IS DOUBLE SHADED 1) y < 2x – 3 2) y > x – 2 Y -x + 2 y < 2x + 1

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3) x 2 4) x 5 y -1 y x + 2

5) x -2 6) y 2

3 x - 1

y 43 x – 2 y 2

3 x + 2

7) x – y 1 8) x – 4y 4 y - ¼ x + 2 2x + y 5

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