Solve equations with variables on both sides. | Linear equations in one variable | SAT Math

Move variable terms to one side, constants to the other, and watch for no-solution or infinitely-many-solution cases.

Core Idea

When variables appear on both sides, move all variable terms to one side and all constants to the other, then solve as usual.

Understanding

The twist here is that subtracting a variable term from one side doesn't eliminate the variable — it moves it. In $7 𝑥 + 2 = 3 𝑥 + 18$ , subtract $3 𝑥$ from both sides to get $4 𝑥 + 2 = 18$ , then solve normally.

Choose which side to keep the variable on strategically. Moving the smaller variable term avoids negatives: in $2 𝑥 + 5 = 8 𝑥 - 7$ , subtract $2 𝑥$ (not $8 𝑥$ ) so you get $5 = 6 𝑥 - 7$ instead of $- 6 𝑥 + 5 = - 7$ .

Watch for special cases. If the variable terms cancel and you get a true statement like $0 = 0$ , the equation has infinitely many solutions. If you get a false statement like $5 = 3$ , there is no solution.

Step by Step

Simplify each side independently (distribute, combine like terms).
Subtract the smaller variable term from both sides to collect variables on one side.
Move constant terms to the opposite side using addition or subtraction.
Divide by the coefficient to isolate the variable.

Misconceptions

Subtracting the variable term from only one side and losing the balance.
Not recognizing a 'no solution' result — when variables cancel and you're left with a false statement, some students keep trying to solve.
Confusing 'no solution' (false statement like $3 = 7$ ) with $𝑥 = 0$ — these are completely different outcomes.

Question

Worked Example

What value of $𝑥$ satisfies $5 𝑥 - 3 = 2 𝑥 + 12$ ?

Select an answer to see the explanation