Systems of Equations

Substitution Method

Solve a system of two equations by substituting one equation into the other. Express one variable in terms of the other, substitute, and solve for the remaining variable, then back-substitute to find both values.

substitution Substitution: isolate one variable in one equation, substitute that expression into the other equation. Solve the single-variable equation, then back-substitute to find the second variable. Check both originals.

Example

Substitute one equation into the other, solve, then back-substitute.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

y=2x−1, 3x+y=9y= 2x - 1 ,\ 3x +y= 9
Use the solved equation to replace y in the other equation.solved equationy = 2x - 1replacement slot3x + y = 9replace ysubstitute--combine x--add 1--solve x--back-substitute--Check: 3(2) + 3 = 9; y = 2x - 1

Step 2 — Substitute

Substitute y = 2x - 1 into 3x + y = 9.

3x+(2x−1)=93x +( \hlmath{2x} - \hl{1} )= 9
replace y: 3x + (2x - 1) = 9.solved equationy = 2x - 1replacement slot3x + y = 9replace ysubstitute3x + (2x - 1) = 9combine x--add 1--solve x--back-substitute--Check: 3(2) + 3 = 9; y = 2x - 1

Step 3 — Combine x terms

Combine x terms: 5x - 1 = 9.

5x−1=9\hlmath{5x} - 1 = 9
Combine x terms: 3x + 2x becomes 5x.solved equationy = 2x - 1replacement slot3x + y = 9replace ysubstitute3x + (2x - 1) = 9combine x5x - 1 = 9add 1--solve x--back-substitute--Check: 3(2) + 3 = 9; y = 2x - 1

Step 4 — Add constant

Add the constant: 9 + 1 = 10.

5x=9+1=105x = 9 + 1 = \hl{10}
Add 1 to both sides.solved equationy = 2x - 1replacement slot3x + y = 9replace ysubstitute3x + (2x - 1) = 9combine x5x - 1 = 9add 15x = 9 + 1 = 10solve x--back-substitute--Check: 3(2) + 3 = 9; y = 2x - 1

Step 5 — Solve x

Solve x: 10 / 5 = 2.

x=10/5=2x= 10 / 5 = \hl{2}
Divide by 5 to find x.solved equationy = 2x - 1replacement slot3x + y = 9replace ysubstitute3x + (2x - 1) = 9combine x5x - 1 = 9add 15x = 9 + 1 = 10solve xx = 10 / 5 = 2back-substitute--Check: 3(2) + 3 = 9; y = 2x - 1

Step 6 — Back-substitute

Back-substitute: 2(2) - 1 = 3.

y=2(2)−1=3y= 2 ( 2 )- 1 = \hl{3}
Put x back into the solved equation to find y.solved equationy = 2x - 1replacement slot3x + y = 9replace ysubstitute3x + (2x - 1) = 9combine x5x - 1 = 9add 15x = 9 + 1 = 10solve xx = 10 / 5 = 2back-substitutey = 2(2) - 1 = 3Check: 3(2) + 3 = 9; y = 2x - 1

Step 7 — Solution

Read the final result.

(2,3)( 2 , 3 )
Solution point: (2, 3).solved equationy = 2x - 1replacement slot3x + y = 9replace ysubstitute3x + (2x - 1) = 9combine x5x - 1 = 9add 15x = 9 + 1 = 10solve xx = 10 / 5 = 2back-substitutey = 2(2) - 1 = 3(2, 3)both lines meetCheck: 3(2) + 3 = 9; y = 2x - 1