Equations with variables on both sides
An equation with variables on both sides, such as 3x + 5 = x + 13, is solved by collecting all variable terms on one side and all constants on the other. Subtracting x and then 5 from both sides gives 2x = 8, so x = 4.
An equation with variables on both sides has the unknown appearing on each side of the equals sign, like 3x + 5 = x + 13. The strategy is the same as for any linear equation — do the same operation to both sides — but with one extra step: first gather all the variable terms together on a single side.
Work it in stages. Starting with 3x + 5 = x + 13, subtract x from both sides to get 2x + 5 = 13. Then subtract 5 from both sides: 2x = 8. Finally, divide both sides by 2 to find x = 4. Check by substituting back: 3(4) + 5 = 17 and 4 + 13 = 17. If the equation contains parentheses, distribute first; if it contains fractions, multiplying every term by the common denominator clears them before you collect terms.
Two special cases are worth recognizing. If the variables cancel completely and leave a false statement — 2x + 3 = 2x + 5 becomes 3 = 5 — the equation has no solution. If they cancel and leave a true statement, like 4 = 4, the equation is an identity and every value of x is a solution.
The ACT math section tests this skill directly and buries it inside word problems and systems of equations, where setting two expressions equal to each other produces exactly this kind of equation. Being fast and accurate at collecting terms is foundational for nearly every algebra question on the exam.
Key takeaways
- Collect all variable terms on one side and all constants on the other, then divide by the coefficient.
- Distribute across parentheses and clear fractions before collecting terms.
- If the variables cancel and leave a false statement, there is no solution; a true statement means infinitely many solutions.
- Always check the answer by substituting it back into the original equation.
- The ACT tests this skill directly and inside word problems and systems of equations.