How to solve linear equations on the ACT

To solve a linear equation, undo what has been done to the variable until it stands alone: distribute, combine like terms, clear fractions, then gather variable terms on one side and numbers on the other. That single routine handles almost every linear equation the ACT can throw at you, and once it is automatic, these become some of the fastest points on the section.

What makes an equation linear

A linear equation has its variables raised only to the first power. No x^2, no variable under a square root, no variable stuck in a denominator. That restriction is a gift: a linear equation in one variable has at most one solution, and you can always reach it by undoing operations in reverse order. Algebra makes up 17 to 20 percent of ACT Math, and linear equations are the spine of it, so see how much algebra is on the ACT for the bigger picture.

Isolate the variable one step at a time

Whatever you do to one side, do to the other. Work outside in: peel off addition and subtraction first, then multiplication and division.

Example: solve 3x + 5 = 20.

• Subtract 5 from both sides: 3x = 15.
• Divide both sides by 3: x = 5.
• Check: 3 times 5 plus 5 is 20. It works.

When the variable shows up on both sides, move it to whichever side keeps the coefficient positive. Solve 7x - 4 = 3x + 16.

• Subtract 3x from both sides: 4x - 4 = 16.
• Add 4 to both sides: 4x = 20.
• Divide by 4: x = 5.
• Check: 7 times 5 minus 4 is 31, and 3 times 5 plus 16 is 31.

Distribute, then combine like terms

Parentheses come first, and this is where careless points disappear. Solve 4(x - 3) - 2(x + 1) = 10.

• Distribute both: 4x - 12 - 2x - 2 = 10.
• Combine like terms: 2x - 14 = 10.
• Add 14: 2x = 24, so x = 12.
• Check: 4 times 9 is 36, 2 times 13 is 26, and 36 minus 26 is 10.

Notice that the minus 2 multiplied both terms inside its parentheses. That is the step students rush.

Clear fractions by multiplying through

Fractions look worse than they are. Multiply every single term by the least common denominator and they vanish. Solve x/3 + x/4 = 14.

• The least common denominator is 12.
• Multiply every term by 12: 4x + 3x = 168.
• Combine: 7x = 168, so x = 24.
• Check: 24 divided by 3 is 8, 24 divided by 4 is 6, and 8 plus 6 is 14.

The word every is doing real work there. Multiplying the fractions but forgetting the 14 is one of the most common ways this problem goes wrong.

Solving for a variable in terms of others

The ACT loves literal equations, where your answer is an expression rather than a number. The moves are identical; you just stop when the target variable is alone.

Solve 3x + 4y = 12 for y. Subtract 3x to get 4y = 12 - 3x, then divide everything by 4 to get y = 3 - (3/4)x. That is now in y = mx + b form, with slope negative 3/4 and y-intercept 3.

Solve P = 2L + 2W for W. Subtract 2L to get P - 2L = 2W, then divide by 2: W = (P - 2L)/2. When the answer choices are expressions, this is the whole question.

Traps the ACT sets

• Sign errors when distributing a negative. Remember that -2(x + 1) is -2x - 2, never -2x + 2.
• Answering for x when the question wants something else. If 5x - 7 = 2x + 11, then 3x = 18, so x = 6. But if the question asks for 3x, the answer is 18, and 6 will be sitting right there as a choice.
• Multiplying only some terms when clearing fractions or distributing.
• Skipping the ten-second check. Substituting your answer back into the original equation catches nearly every arithmetic slip.

These overlap with the broader list in the most common ACT Math mistakes, and they are worth knowing by name so you can catch yourself.

A quick self-check

Try these three, then compare:

• 6x - 9 = 2x + 15
• 5(2x - 1) = 3x + 9
• Solve C = 2(pi)r for r.

Answers: x = 6, x = 2, and r = C/(2 pi). If any of those felt slow, the fix is volume, not theory. Run a set of concept quizzes on thirty-six until the steps happen without narration. Then move on to solving systems of equations, which is really just this same skill applied twice.

This article offers general ACT prep guidance. The ACT can change from year to year, including its format, scoring, policies, test dates, and fees, so always confirm the latest details on the official ACT website at act.org before you make decisions. ACT® is a registered trademark of ACT, Inc. thirty-six is not affiliated with, endorsed by, or sponsored by ACT.