# 8-3a Simultaneous Solutions - variation for Addition method

Notes here, and on pages 365-368 in book

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1) Align x's over x's and y's over y's. Will likely require some movement via the commutative property or adding of opposites to get variables in the correct spots.

1b) Create opposite terms: You may need to multiply one or both equations by -1 or some other number in order for the x's or the y's to be opposites.

2) Add If 1 (and 1b) are done properly, one of the variables will be eliminated (why this is sometimes called the elimination method).

3) Solve the new equation which now has only one variable

4) Use the value of the variable to plug in to one of the original equations to find the missing side of the coordinate pair.

Example:

For the equations, x-y=2 and y+3x=10
What are the coordinates of the simultaneous solution?

ALIGN: In this case just move around the first equation to make it -y+x=2

y+3x=10

-y+x=2

Gives: 4x=12

SOLVE: 4x=12 this means x=3

PLUG IN: Pick either of the two equations to plug in x=3. You should get the same answer for y.

-y+3=2

gives

y = 1

whereas

y+3(3)=10

also gives

y=1

So the solution is (4,1)

Harder Example:

For the equations, x+y=1 and 3y-7=-4x
What are the coordinates of the simultaneous solution?

ALIGN: In this case just move around the second equation to make it 4x+3y=7

4x+3y=7

x+y=1

DOES NOT ELIMINATE X or Y

MUST DO STEP 1B and Multiply second equation by -4 (or -3) to create opposite terms

4x+3y=7

-4x-4y=-4

Gives: -y=3

SOLVE: -y=3 this means y=-3

PLUG IN: Pick either of the two equations to plug in y=-3.

x+-3=1

x=4

So the solution is (4,-3)