Warm up 12/6 or 7 Warm up 12/6 or 7 1)1) Write the equation of a line that is Write the equation of a line that is

parallel to y = -3x –5 and goes parallel to y = -3x –5 and goes through the point (6,10).through the point (6,10).

2)2) Write the equation of a line that is Write the equation of a line that is perpendicular to y = -3x –5 and perpendicular to y = -3x –5 and goes through the point (6,10).goes through the point (6,10).

3) Using elimination, solve the 3) Using elimination, solve the following system. following system.

a) x + y = 4 -3x + y = -8a) x + y = 4 -3x + y = -8

b) 3x – 4y = -15 5x + y b) 3x – 4y = -15 5x + y = -2= -2

Answers to warm upAnswers to warm up1) y = -3x + 28

2) Y = 1/3x +8

3) A) Subtract the two lines : x + y = 4

-3x + y = -8

4x = 12 x = 3 Substitute: y = 1

(3, 1)

b) (-1, 3)

Solving Systems by Solving Systems by Elimination cont.Elimination cont.

Objective: To find the solution Objective: To find the solution to a system of equations by to a system of equations by elimination (addition).elimination (addition).Remember: The solution is the point Remember: The solution is the point where the lines intersect.where the lines intersect.

1) 2x + 3y = 61) 2x + 3y = 6

+5x – 4y = -8+5x – 4y = -8

4(2x + 3y) = 6(4) 8x + 12y = 244(2x + 3y) = 6(4) 8x + 12y = 24

3(5x – 4y) = -8(3)3(5x – 4y) = -8(3) +15x -12y = -24+15x -12y = -24

23 x = 023 x = 0

x = 0x = 0Now plug 0 in for x into any of the 4 equations.Now plug 0 in for x into any of the 4 equations.

2(0) + 3y = 62(0) + 3y = 6

3y = 63y = 6

Y = 2Y = 2

We will need to change both equations. We will have the y value drop out.

The solution is (0, 2)

You Try:You Try:

1.1. 2x + 3y = 142x + 3y = 14

3x – 2y = -53x – 2y = -5

2.2. 5x + 2y = -85x + 2y = -8

2x – 5y = -92x – 5y = -9

(1, 4)

(-2, 1)

Solve Systems by EliminationSolve Systems by EliminationThis time both equations will be multiplied but one This time both equations will be multiplied but one must be multiplied by opposite sign to be canceled must be multiplied by opposite sign to be canceled out.out.Example 2) 4x+ 2y = -8Example 2) 4x+ 2y = -8

5x + 7y = 85x + 7y = 8

Step 1 – Choose one variable to eliminate. Let’s start with x.Step 1 – Choose one variable to eliminate. Let’s start with x.

One line will need to be multiplied by a negative number since One line will need to be multiplied by a negative number since all coefficients are positive.all coefficients are positive.

Let’s choose the first line.Let’s choose the first line.

-5(4x+ 2y) = -8(-5)-5(4x+ 2y) = -8(-5)

4(5x + 7y = 8(4)4(5x + 7y = 8(4)

-20x -10y = 40-20x -10y = 40

20x + 28y = 3220x + 28y = 32

18y = 7218y = 72

y=4y=4

Now substitute y back in to any of the four equations find x.Now substitute y back in to any of the four equations find x.

Example 2 Continued : Example 2 Continued : 4x+ 2y = -84x+ 2y = -8 5x + 7y = 85x + 7y = 8

4x + 2(4) = -84x + 2(4) = -8

4x + 8 = -84x + 8 = -8

-8-8 -8 -8

4x = -164x = -16

X = -4X = -4

The solution is (-4,4)The solution is (-4,4)

YOU TRY!!YOU TRY!!

8x+ 2 y = 08x+ 2 y = 0

5x + 3y = -75x + 3y = -7

Answer: (1,-4)Answer: (1,-4)

Summary: What is the solution Summary: What is the solution of a system of linear of a system of linear equations?equations?Homework:7.4B WSHomework:7.4B WS