EQN-03-01

Solving Linear Equations

Isolate the variable using inverse operations — the single most reused algebra skill on the GAT-E.

~5 min read•Difficulty 2/5

An equation is a balance. Whatever you do to one side, do to the other. Your goal is to get the variable alone on one side.

Undo each operation with its opposite, in reverse PEMDAS order:

3 x + 5 = 20

3 x = 15 (subtract 5 from both sides)

x = 5 (divide both sides by 3)

Move all variable terms to one side, all constants to the other.

5 x - 4 = 2 x + 11

3 x = 15 (subtract 2 x, add 4)

x = 5

Multiply the whole equation by the common denominator first — it removes the fractions in one step.

x /2 + x /3 = 5 \to \times 6 \to 3 x + 2 x = 30 \to 5 x = 30 \to x = 6

Substitute your answer back into the original equation. On the GAT-E this catches almost every sign error in seconds.

Worked examples

Example 1 (difficulty 1/5)

Solve

: 4 x - 7 = 13

◎ Predict the answer first:

Example 2 (difficulty 2/5)

Solve

: 2 (x + 3) = x + 11

◎ Predict the answer first:

When the answers are simple numbers, plugging each choice back in ("backsolving") can be faster than solving.
Clear fractions and parentheses before collecting like terms.
Keep the equals sign aligned down the page — it prevents losing a term.

Distributing to only the first term

: 2 (x + 3) = 2 x + 3

Why: The second term inside the parentheses is overlooked.

Fix: Multiply the outside factor by every term inside

: 2 x + 6.

Dividing only part of a side by the coefficient

Why: Rushing the last step.

Fix: Divide the entire side (every term) by the coefficient.