Method guideAlgebra IFoundational9 min read

Solving linear equations: keep the balance, not a bag of tricks

An equation stays true when the same legal operation is applied to both sides. Simplify, isolate, and check.

3(x-2)+5=2x+9

Method guide

How to recognize the method, run it, and know when it is the wrong choice.

Updated July 13, 2026

GoalStart here

ax+b=c\quad\Longrightarrow\quad x=\frac{c-b}{a}

The compact formula is useful, but the reliable habit is balance: perform the same reversible move on both sides.

Start by reducing clutter

Distribute and combine like terms on each side before moving terms across the equals sign. This exposes the equation's real structure.

Fractions can often be cleared by multiplying every term on both sides by a common denominator. Skipping a term breaks the balance.

\frac{x}{3}+\frac12=2\quad\Longrightarrow\quad2x+3=12

Move variable terms to one side and constants to the other. Then undo multiplication or division around the variable.

There is rarely one mandatory sequence. Prefer moves that keep coefficients small and reduce the chance of sign errors.

5x-7=2x+8\quad\Longrightarrow\quad3x=15

Substitute the proposed value into the original equation, not a later simplified line. Both sides should produce the same number.

A check catches arithmetic slips and also reveals equations with no solution or infinitely many solutions when the variable disappears.

x=5:\quad5(5)-7=2(5)+8=18

Worked exampleBalance both sides

3x-6+5=2x+9

Distribute the three.

3x-1=2x+9

Combine constants on the left.

x-1=9

Subtract 2x from both sides.

x=10

Add one to both sides.

Result

\boxed{x=10}

Watch for

Keep