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Linear Equations for the PSAT/NMSQT: Complete Study Guide + Free Practice Problems
Everything the PSAT/NMSQT tests about linear equations in one place: solving one-step and multi-step equations, variables on both sides, equations with fractions and decimals, literal equations, and word problems — with step-by-step examples, worked problems, and 2 free practice quizzes.
Practice these quizzes
Free · No signupWhat is a linear equation?
A linear equation is an equation where the variable appears only to the first power — no squares, no roots, no variable in a denominator. Solving one means isolating the variable on one side using inverse operations, always keeping both sides of the equation balanced.
Whatever operation you perform, apply it to both sides of the equation at the same time. Treating the equals sign as a balance point is the single most important habit for solving equations without errors.
Solving multi-step equations
Solve for x: 4(x − 3) + 5 = 21
Distribute and combine like terms before doing anything else. Simplifying each side fully first turns a messy multi-step equation into a simple two-step equation.
Variables on both sides
Solve for x: 5x + 9 = 2x + 27
Moving a variable term to the other side always changes its sign, just like moving a constant. Forgetting to flip the sign when a variable term crosses the equals sign is a frequent source of errors.
Equations with fractions & decimals
The fastest way to solve an equation containing fractions or decimals is to eliminate them in the very first step, turning the equation into simple whole numbers before solving.
Solve for x: x/3 + x/4 = 7
Multiply every single term on both sides of the equation by the common denominator — including any term that isn't already a fraction. Missing one term is the most common way this shortcut goes wrong.
Literal equations
A literal equation contains multiple variables, and you're asked to solve for just one of them in terms of the others — treating every other letter as if it were a constant.
Solve the equation A = ½bh for h.
Treat a literal equation exactly like a normal equation — the only difference is that the "numbers" surrounding your target variable happen to be letters instead. The same inverse operations still apply.
Linear equation word problems
Many PSAT questions describe a situation in words and expect you to translate it into an equation before solving. Define a variable clearly, then translate each phrase piece by piece.
Three more than twice a number is 25. Find the number.
Common PSAT traps
- Distributing a negative incorrectly: −(2x − 5) equals −2x + 5, not −2x − 5.
- Forgetting to apply an operation to every term: when clearing fractions or distributing, every single term must be affected, not just the first one.
- Sign errors when moving variable terms: a term always changes sign when it crosses the equals sign, whether it's a constant or a variable term.
- Solving for the wrong variable in a literal equation: double-check the question is asking to isolate before finalizing an answer.
Test day strategy for linear equations
| Question signal | Fastest approach |
|---|---|
| Parentheses in the equation | Distribute first, then combine like terms before isolating the variable |
| Variable terms on both sides | Move all variable terms to one side first, constants to the other |
| Fractions or decimals present | Multiply every term by the common denominator (or power of 10) to clear them |
| Multiple variables, solve for one | Treat other variables as constants; isolate the target variable last |
| Word problem with no equation given | Define a variable explicitly, then translate the sentence piece by piece |
Now put it to work
Two quiz sets, each building on the last — start with Quiz 1 and work through in order, or jump straight to the topic you need.
Linear Equations 1
Multi-step equations, variables on both sides, fractions and decimals.
Start quiz → Quiz 2Linear Equations 2
Literal equations and word problems.
Start quiz →