PSAT/NMSQT Math: Linear Equations
Linear equations are the foundation of the Heart of Algebra content on the PSAT/NMSQT, and they show up constantly, both as standalone problems and buried inside harder questions on functions, systems, and word problems. This free, complete lesson covers one-step and multi-step equations, the distributive property, equations with fractions, variables on both sides, and how to translate word problems into equations you can solve. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback and full explanations. Everything here is free, and you can keep practicing afterward with the linked quizzes below.
1. One-Step and Multi-Step Equations
A linear equation in one variable can always be solved by isolating that variable using inverse operations. Whatever you do to one side of the equation, you must do to the other. A one-step equation needs a single inverse operation; a multi-step equation needs several, usually undoing addition or subtraction before undoing multiplication or division.
Worked Example: Solve 4x − 7 = 21.
4x − 7 = 21 → 4x = 28 → x = 7
1. Solve for x: 3x + 5 = 26
Explanation: 3x + 5 = 26 → 3x = 21 → x = 7.
2. Solve for y: y4 − 3 = 2
Explanation: Add 3 to both sides: y/4 = 5. Multiply both sides by 4: y = 20.
3. Solve for n: −2n + 9 = −13
Explanation: −2n + 9 = −13 → −2n = −22 → n = 11. Dividing by a negative number does not change the direction of an equation (only inequalities), so the sign here just tracks through the arithmetic.
4. Solve for x: 5(x − 2) = 3x + 4, given that this is really testing the distributive property as a first step.
Explanation: 5x − 10 = 3x + 4 → 2x = 14 → x = 7.
2. The Distributive Property and Equations with Fractions
Many PSAT equations are written so that you must first distribute a number across parentheses, or clear fractions by multiplying every term by a common denominator, before the equation looks like the simple one-step or multi-step forms above. Clearing fractions early almost always makes an equation easier and less error-prone to solve.
Worked Example: Solve x3 + x2 = 10.
Multiply every term by 6 (the LCD of 3 and 2):
2x + 3x = 60 → 5x = 60 → x = 12
1. Solve for x: 3(2x + 1) = 21
Explanation: 6x + 3 = 21 → 6x = 18 → x = 3.
2. Solve for x: x + 24 = 5
Explanation: Multiply both sides by 4: x + 2 = 20, so x = 18.
3. Solve for x: x2 − x5 = 3
Explanation: Multiply every term by 10 (the LCD of 2 and 5): 5x − 2x = 30 → 3x = 30 → x = 10.
4. Solve for x: −3(x − 4) + 2 = 5x
Explanation: −3x + 12 + 2 = 5x → −3x + 14 = 5x → 14 = 8x → x = 14/8 = 7/4.
3. Variables on Both Sides
When a variable appears on both sides of an equation, gather all variable terms onto one side and all constants onto the other before isolating the variable. It doesn't matter which side you move the variable to, but choosing to move it toward the side that leaves a positive coefficient usually avoids extra sign errors.
Worked Example: Solve 7x − 4 = 2x + 16.
Subtract 2x from both sides: 5x − 4 = 16.
5x = 20 → x = 4
1. Solve for x: 6x + 3 = 2x + 19
Explanation: 4x + 3 = 19 → 4x = 16 → x = 4.
2. Solve for x: 4x − 9 = 9x + 6
Explanation: −9 − 6 = 9x − 4x → −15 = 5x → x = −3.
3. Solve for x: 2(x + 5) = 3(x − 1)
Explanation: 2x + 10 = 3x − 3 → 10 + 3 = 3x − 2x → x = 13.
4. Which equation has no solution?
Explanation: Moving the variable terms in A gives 5 = 9, a false statement, so no value of x works. (B is true for every x, and C and D each simplify to a single valid solution.)
4. Writing Equations from Word Problems
A large share of PSAT linear equation questions describe a real-world situation in words and ask you to write, interpret, or solve the matching equation. Read carefully to identify what the unknown quantity represents, translate the relationships in order, and pay close attention to what each coefficient and constant actually means in context, since some questions only ask you to interpret a piece of the equation rather than solve it.
Worked Example: A rectangular garden has a perimeter of 54 feet. Its length is 3 feet more than twice its width. Write an equation for the width w, and solve for w.
Length = 2w + 3. Perimeter: 2(length) + 2(width) = 54.
2(2w + 3) + 2w = 54 → 4w + 6 + 2w = 54 → 6w = 48 → w = 8
1. A phone plan costs $30 per month plus $0.10 per text message sent beyond the included limit. If a customer's bill for one month was $42, how many texts beyond the limit, t, did they send?
Explanation: 30 + 0.10t = 42 → 0.10t = 12 → t = 120.
2. The equation C = 45h + 60 models the total cost, in dollars, of hiring a plumber for h hours. What does the 45 represent in this context?
Explanation: Since 45 is multiplied by h (hours), it represents the rate charged per hour; the 60 is the flat fee added regardless of hours.
3. Three consecutive integers have a sum of 84. What is the smallest of the three integers?
Explanation: Let the integers be n, n+1, n+2. Then 3n + 3 = 84 → 3n = 81 → n = 27, the smallest integer.
4. A rectangle's perimeter is 54 feet, and its length is 3 feet more than twice its width w. Which equation correctly represents this situation?
Explanation: Length = 2w + 3, and perimeter = 2(length) + 2(width), giving 2(2w + 3) + 2w = 54.
5. Using the equation from the previous problem, 2(2w + 3) + 2w = 54, what is the width w?
Explanation: 4w + 6 + 2w = 54 → 6w = 48 → w = 8 feet.
Common Mistakes to Avoid
- Forgetting to distribute to every term inside parentheses. In an expression like 5(x − 2), both x and 2 must be multiplied by 5, not just the first term.
- Dropping a negative sign when moving a term across the equal sign. Moving a term always flips its sign; track this carefully, especially with subtraction.
- Multiplying only some terms by the common denominator when clearing fractions. Every single term on both sides of the equation, including any lone constant, must be multiplied.
- Misreading what a coefficient or constant represents in a word problem. Some questions only ask you to interpret part of an equation, not solve it, so always re-read exactly what is being asked before doing extra work.
- Assuming every equation has exactly one solution. An equation can also have no solution (a false statement after simplifying) or infinitely many solutions (a true statement for every value), both of which appear on the PSAT.
Frequently Asked Questions
What's the difference between a one-step and a multi-step equation?
A one-step equation can be solved with a single inverse operation, such as adding, subtracting, multiplying, or dividing once. A multi-step equation requires more than one operation, and often needs the distributive property or fraction-clearing applied first before it reduces to a simple, solvable form.
How do I know which side to move the variable to when it appears on both sides?
Either side works mathematically, but moving the variable toward whichever side keeps its coefficient positive after combining terms tends to avoid extra sign mistakes.
How can I tell if a word problem needs an equation with variables on both sides?
Look for two separate quantities that are being set equal to each other, such as two different pricing plans reaching the same total cost, rather than a single expression being set equal to one known number.
Where can I practice more problems like these?
The Linear Equations quizzes in the PSAT/NMSQT Math Question Bank include additional original problems on this topic, along with quizzes covering every other Heart of Algebra, Advanced Math, Problem Solving and Data Analysis, and Geometry and Trigonometry skill tested on the PSAT/NMSQT. You can also browse the full PSAT/NMSQT Math Question Bank.