Linear Equations

Variables on Both Sides

Solve a linear equation where the variable appears on both sides by collecting variable terms on one side.

Example

Move variable terms to one side, then solve.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

5x−3=2x+95x - 3 = 2x + 9
Start with x terms on both sides.left side5x - 3=right side2x + 9both sidesmove x terms to one side--then undo -3--divide by 3resultStart: 5x - 3 = 2x + 9

Step 2 — Move x terms

Subtract 2x from both sides to move the 2x term.

5x−2x−3=2x−2x+95x - \hlmath{2x} - 3 = 2x - \hlmath{2x} + 9
Subtract 2x from both sides.left side5x - 3=right side2x + 9both sidessubtract 2x: 5x - 2x = 3x--then undo -3--divide by 3resultStart: 5x - 3 = 2x + 9

Step 3 — Simplify variable side

Collect x terms: 3x - 3 = 9.

3x−3=9\hlmath{3x} - 3 = 9
Collect x terms: 3x - 3 = 9.left side5x - 3=right side2x + 9both sidessubtract 2x: 5x - 2x = 3x3x - 3 = 9then undo -3--divide by 3resultStart: 5x - 3 = 2x + 9

Step 4 — Add constant

Add the constant: 9 + 3 = 12.

3x=9+3=123x = 9 + 3 = \hl{12}
Add 3 to both sides: 3x = 12.left side5x - 3=right side2x + 9both sidessubtract 2x: 5x - 2x = 3x3x - 3 = 9add 3: 9 + 3 = 123x = 12divide by 3resultStart: 5x - 3 = 2x + 9

Step 5 — Divide

Divide to solve: 12 / 3 = 4.

x=12/3=4x = 12 / 3 = \hl{4}
Divide by 3: 12 / 3 = 4.left side5x - 3=right side2x + 9both sidessubtract 2x: 5x - 2x = 3x3x - 3 = 9add 3: 9 + 3 = 123x = 1212 / 3 = 4resultStart: 5x - 3 = 2x + 9

Step 6 — Result

Read the final result.

x=4x = 4
Result: x = 4.left side5x - 3=right side2x + 9both sidessubtract 2x: 5x - 2x = 3x3x - 3 = 9add 3: 9 + 3 = 123x = 1212 / 3 = 4x = 4Start: 5x - 3 = 2x + 9
variables-both-sides Subtract the smaller variable term from both sides to collect x, then solve the resulting two-step equation.