Systems of Equations

Elimination Method

Solve a system of two equations by adding them to eliminate one variable. Choose equations whose opposite coefficients cancel. After finding one variable, back-substitute into either original equation to find the other.

elimination Elimination: add (or subtract) equations so one variable cancels. Solve the resulting single-variable equation. Back-substitute to find the second variable. Verify the solution satisfies both originals.

Example

Add equations to eliminate one variable, then back-substitute.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

2x+3y=12, x−3y=−32x + 3y = 12 ,\ x- 3y =- 3
Stack the equations so opposite y terms line up.equation 12x + 3y = 12equation 2x - 3y = -3add equationscancel y--remaining--solve x--back-substitute--isolate y--solve y--Check: 2x + 3y = 12 and x - 3y = -3 share (3, 2)

Step 2 — Add equations

Add equations so y cancels: 3y + (-3)y = 0.

3y+(−3y)=0\hlmath{3y} +(- \hlmath{3y} )= \hl{0}
Add equations so the y terms cancel.equation 12x + 3y = 12equation 2x - 3y = -3add equationscancel y3y + (-3)y = 0remaining--solve x--back-substitute--isolate y--solve y--Check: 2x + 3y = 12 and x - 3y = -3 share (3, 2)

Step 3 — Eliminate y

After eliminating y: 3x = 9.

3x=9\hlmath{3x} = \hl{9}
Only the x equation remains.equation 12x + 3y = 12equation 2x - 3y = -3add equationscancel y3y + (-3)y = 0remaining3x = 9solve x--back-substitute--isolate y--solve y--Check: 2x + 3y = 12 and x - 3y = -3 share (3, 2)

Step 4 — Solve x

Solve x: 9 / 3 = 3.

x=9/3=3x= 9 / 3 = \hl{3}
Divide to find x.equation 12x + 3y = 12equation 2x - 3y = -3add equationscancel y3y + (-3)y = 0remaining3x = 9solve xx = 9 / 3 = 3back-substitute--isolate y--solve y--Check: 2x + 3y = 12 and x - 3y = -3 share (3, 2)

Step 5 — Back-substitute

Back-substitute into the first equation: 2(3) + 3y = 12.

2(3)+3y=122 ( \hl{3} )+ 3y = 12
Put x back into the first equation.equation 12x + 3y = 12equation 2x - 3y = -3add equationscancel y3y + (-3)y = 0remaining3x = 9solve xx = 9 / 3 = 3back-substitute2(3) + 3y = 12isolate y--solve y--Check: 2x + 3y = 12 and x - 3y = -3 share (3, 2)

Step 6 — Isolate y term

Subtract the x part: 12 - 6 = 6.

3y=12−6=63y = 12 - 6 = \hl{6}
Subtract the x part to isolate the y term.equation 12x + 3y = 12equation 2x - 3y = -3add equationscancel y3y + (-3)y = 0remaining3x = 9solve xx = 9 / 3 = 3back-substitute2(3) + 3y = 12isolate y3y = 12 - 6 = 6solve y--Check: 2x + 3y = 12 and x - 3y = -3 share (3, 2)

Step 7 — Solve y

Solve y: 6 / 3 = 2.

y=6/3=2y= 6 / 3 = \hl{2}
Divide to find y.equation 12x + 3y = 12equation 2x - 3y = -3add equationscancel y3y + (-3)y = 0remaining3x = 9solve xx = 9 / 3 = 3back-substitute2(3) + 3y = 12isolate y3y = 12 - 6 = 6solve yy = 6 / 3 = 2Check: 2x + 3y = 12 and x - 3y = -3 share (3, 2)

Step 8 — Solution

Read the final result.

(3,2)( 3 , 2 )
Solution point: (3, 2).equation 12x + 3y = 12equation 2x - 3y = -3add equationscancel y3y + (-3)y = 0remaining3x = 9solve xx = 9 / 3 = 3back-substitute2(3) + 3y = 12isolate y3y = 12 - 6 = 6solve yy = 6 / 3 = 2(3, 2)both equations meetCheck: 2x + 3y = 12 and x - 3y = -3 share (3, 2)