Lesson 7.4ALesson 7.4A

Solving Linear Systems Using Solving Linear Systems Using EliminationElimination

Keys to KnowKeys to Know

Solving with Elimination:Solving with Elimination:

When you combine the equations to When you combine the equations to get rid of (eliminate) one of the get rid of (eliminate) one of the variables.variables.

Possible solutions are:Possible solutions are:

One SolutionOne Solution

No SolutionsNo Solutions

Infinite SolutionsInfinite Solutions

Steps for Using Steps for Using EliminationElimination

1)1) Write both equations in standard form Write both equations in standard form (Ax + By = C) so that variables and = line up(Ax + By = C) so that variables and = line up

2)2) Multiply one or both equations by a number Multiply one or both equations by a number to make opposite coefficients on one to make opposite coefficients on one variable.variable.

3)3) Add equations together (one variable should Add equations together (one variable should cancel out)cancel out)

4)4) Solve for remaining variable.Solve for remaining variable.5)5) Substitute the solution back in to find other Substitute the solution back in to find other

variable.variable.6)6) Write the solution as an ordered pairWrite the solution as an ordered pair7)7) Check your answerCheck your answer

Example 1:Example 1: 5x + y = 125x + y = 12 3x – y = 43x – y = 4 8x = 168x = 16 8 88 8 x = 2x = 2

5(2) + y = 125(2) + y = 1210 + y = 1210 + y = 12y = 2y = 2

The solution is: (2, 2)The solution is: (2, 2)

Step 1: Put both equations in standard form.

Step 2: Check for opposite coefficients.

Step 3: Add equations together

Step 4: Solve for x

Step 5: Substitute 2 in for x to solve for y (in either equation)

Already Done

y and –y are already opposites

Your TurnYour Turn

Ex. 2 2x + y = 0Ex. 2 2x + y = 0

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

Answer: (-1, 2)Answer: (-1, 2)

Example 3Example 3 3x + 5y = 103x + 5y = 10 3x + y = 23x + y = 2

3x + 5y = 103x + 5y = 10 -1(3x + y) = -1(2-1(3x + y) = -1(2))

4y = 84y = 8 y = 2y = 2Now plug (2) in for y.Now plug (2) in for y.3x + 2 = 23x + 2 = 2X = 0 X = 0 Solution is : (0,2)Solution is : (0,2)

When you add these neither variable drops out

SO….

We need to change 1 or both equations by multiplying the equation by a number that will create opposite coefficients.

When we need to create When we need to create opposite coefficientsopposite coefficients

3x + 5y = 103x + 5y = 10

-3x – y = -2-3x – y = -2

Multiply the bottom equation by negative one to eliminate the x

4) -2x + 3y = 64) -2x + 3y = 6 x – 4y = -8x – 4y = -8

-2x + 3y= 6 -2x + 3y = 6-2x + 3y= 6 -2x + 3y = 62( x – 4y) = -8(2)2( x – 4y) = -8(2) 2x - 8y = -162x - 8y = -16

-5 y = -10-5 y = -10 y = 2y = 2

Now plug (2) in for y into any of the 4 equations.Now plug (2) in for y into any of the 4 equations.-2x + 3(2) = 6-2x + 3(2) = 6-2x + 6 = 6-2x + 6 = 6-2x = 0-2x = 0 x = 0x = 0Solution is: (0, 2)Solution is: (0, 2) Check your work!Check your work!

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

Your TurnYour Turn

Ex. 5Ex. 5 5x – 2y = 125x – 2y = 122x – 2y = -62x – 2y = -6

Ex. 6Ex. 6 -3x + 6y = 9-3x + 6y = 9 x - 2y = -3x - 2y = -3

Ex. 7Ex. 7 2x + 4y = 82x + 4y = 8 x + 2y = 3x + 2y = 3

(6, 9)

0=0Infinite solutions

0=2No Solutions