Let’s Warm Up!1) Solve the system of equations by graphing:

2x + 3y = 12 2x – y = 4

Answer:

2) Find the slope-intercept form for the equation of a line that passes through (0, 5) and is parallel to a line whose equation is 4x – y = 3?

Answer:

3)Solve 3│x – 5│= 12Answer:

(3, 2)

y=4x+5

x= 1, 9

Let’s chat about finals

Wednesday Jan 22nd : 2, 4, 6 Thursday Jan 23rd : 1, 3, 5

Minimum Days

Final Review sheet due DAY OF FINAL

Mini Quiz Time!

3 graphing Questions Get out a pencil please.

8-2 Substitution

Objective: To use the substitution method to solve systems of equations.

Two Algebraic Methods:

Substitution Method Elimination Method will learn

about next

RECALL…Three Types of Solutions: Intersection is Solution

One Solution No Solution

Infinite SolutionsSame slope

Different y-intercept“Run parallel Never

intersect”

Same slope Same y-intercept

“Same line Intersect infinitely”

Different slopeDifferent y-

intercept“Intersect at one

point”

Substitution Method

Use the substitution method when: one equation is set equal to a variable

y = 2x + 1 or x = 3y - 2

Example 1

Instead of x = 2 we have:x = y + 2x + 2y = 11

(y + 2) + 2y = 113y + 2 = 113y = 9y = 3

These are all the same!

x = 3 + 2x = 5

Answer: (5,3)

Try with a Mathlete

1) y = 3x x + 2y = -21

2) y = 2x – 6 3x + 2y = 9

Answers:1) (-3,-9)2) (3,0)

Example 2

x + 4y = 12x – 3y = -9

First, solve for a variablex = -4y + 1

2(-4y + 1) – 3y = -9-8y + 2 – 3y = -9-11y + 2 = -9-11y = -11y = 1

x = -4(1) + 1x = -3

Answer: (-3,1)

Solve for x (because there is no number in front of it)

TOO

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

9

Answers:1) (2,-3)2) (13,30)

Special Cases

x + y = 16 x = 16 – y2y = -2x + 2

2y = -2(16 – y) + 22y = -32 + 2y + 22y = -30 + 2y0 = -30FalseNO SOLUTION

6x – 2y = -4 y = 3x + 2

6x – 2(3x + 2) = -46x – 6x – 4 = -4-4 = -4TrueINFINITELY MANY

TOO for Homework

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

2) x + y = 0, 3x + y = -8

3) y = 3x – 7, 3x – y = 7

Homework

Pg. 467 #17-32 left column

Pg. 467 #17-32 Left ColumnSolve using substitution.

17.

20.

23.

26.

29.

32.

MORE Explanations

The following slides have more examples and explanations of the substitution method.

Examples: Use the substitution method to solve the system of equations.

1) 2x + 3y = 2 x – 3y = –17

x – 3y = –17 +3y +3y x = 3y – 17

2x + 3y = 2 “x = 3y – 17” 2(3y – 17) + 3y = 2 6y – 34 + 3y = 2 –34 + 9y = 2 +34 +34 9y = 36

9 9 y = 4

x = 3y – 17 “y = 4” x = 3(4) – 17 x = 12 – 17 x = –5

1st: Transform one equation to isolate a variable

2nd: Substitute into the other equation and solve for variable #1

3rd: Substitute into transformed equation from 1st step and solve for variable #2

(use the substitution method when a variable is already isolated or when a variable has a coefficient of 1 and can easily be transformed)

Write answer as an ordered pair (x, y): One Solution(–5 , 4)

(we picked x – 3y = – 17 because x has a coefficient of 1 and can easily be transformed)

Examples: Use the substitution method to solve the system of equations.

2) –9x + 3y = –21 3x – y = 7

3x – y = 7-3x -3x –y = –3x + 7 -1 -1 -1 y = 3x – 7

–9x + 3y = –21 “y = 3x – 7” –9x + 3(3x – 7) = –21 –9x + 9x – 21= –21 –21 = –21

1st: Transform one equation to isolate a variable

2nd: Substitute it into the other equation and solve for variable #1

3rd: Substitute into the transformed equation from 1st step and solve for variable #2

(use the substitution method when a variable is already isolated or when a variable has a coefficient of 1 and can easily be transformed)

Write answer as an ordered pair (x, y): Infinite Solutions

(we picked 3x – y = 7 because y has a coefficient of -1 and can easily be transformed)

True!!

Examples: Use the substitution method to solve the system of equations.

3) 4x – 2y = 5 y = 2x + 1

4x – 2y = 5 “y = 2x + 1” 4x – 2(2x + 1) = 5 4x – 4x – 2 = 5 – 2 = 5

1st: Transform one equation to isolate a variable

2nd: Substitute into the other equation and solve for variable #1

3rd: Substitute into transformed equation from 1st step and solve for variable #2

(use the substitution method when a variable is already isolated or when a variable has a coefficient of 1 and can easily be transformed)

No Solution

y = 2x + 1

(already isolated)

False!!

Chapter 8 Systems of Equations

8-2 Substitution

We will become experts at solving systems of equations with substitution.

Math Lab

Solve the system with substitution:x = y + 2x + 2y = 11

(y + 2) + 2y = 113y + 2 = 113y = 9y = 3

These are all the same!

x = 3 + 2x = 5

Answer: (5,3)

Math Lab ReviewSubstitution

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

2) x + y = 0 3x + y = -8

3) y = 3x – 7 3x – y = 7

Special Cases

x + y = 16 x = 16 – y2y = -2x + 2

2y = -2(16 – y) + 22y = -32 + 2y + 22y = -30 + 2y0 = -30FalseNO SOLUTION

6x – 2y = -4 y = 3x + 2

6x – 2(3x + 2) = -46x – 6x – 4 = -4-4 = -4TrueINFINITELY MANY

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