PSAT 8/9 Math: Linear Equations
Linear equations are one of the most frequently tested skills on the PSAT 8/9, and once you're comfortable isolating a variable, almost every version of the question becomes manageable. This free, complete lesson covers variables on both sides, the distributive property and no-solution equations, proportions, and word problems that use linear models. 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 quiz below.
1. Variables on Both Sides
When a linear equation has the variable on both sides, move all variable terms to one side and all constants to the other, then isolate the variable. It helps to move the variable term in whichever direction keeps its coefficient positive.
Worked Example: Solve 5x − 3 = 2x + 12.
Subtract 2x from both sides: 3x − 3 = 12.
Add 3 to both sides: 3x = 15.
Divide by 3: x = 5
1. Solve 5x − 3 = 2x + 12. (see worked example above)
Explanation: Subtract 2x from both sides: 3x − 3 = 12. Add 3: 3x = 15. Divide by 3: x = 5.
2. Solve 7x − 2 = 5x + 14.
Explanation: Subtract 5x from both sides: 2x − 2 = 14. Add 2: 2x = 16. Divide by 2: x = 8.
3. Solve 9x + 4 = 3x + 28.
Explanation: Subtract 3x from both sides: 6x + 4 = 28. Subtract 4: 6x = 24. Divide by 6: x = 4.
2. The Distributive Property and No-Solution Equations
When an equation has a number multiplied by a parenthesis, distribute first, then combine like terms. Sometimes, after fully simplifying, the variable terms cancel out entirely, this means the equation has either no solution (if the remaining constants are unequal) or infinitely many solutions (if they are equal).
Worked Example: Solve 5(x − 2) − 3(x + 4) = 2x + 1.
Distribute: 5x − 10 − 3x − 12 = 2x + 1.
Combine like terms on the left: 2x − 22 = 2x + 1.
Subtract 2x from both sides: −22 = 1, which is never true.
No Solution
1. Solve 5(x − 2) − 3(x + 4) = 2x + 1. (see worked example above)
Explanation: Distributing and combining like terms gives 2x − 22 = 2x + 1. Subtracting 2x from both sides leaves −22 = 1, a false statement, so there is no solution.
2. Simplify (−4x + 5) − (6x + 7) = 0, then solve for x.
Explanation: Distribute the negative sign: −4x + 5 − 6x − 7 = 0. Combine like terms: −10x − 2 = 0. Add 2: −10x = 2. Divide by −10: x = −0.2.
3. Solve 3(2x − 4) = 6x − 12 for x.
Explanation: Distribute: 6x − 12 = 6x − 12. This is true for every value of x, so there are infinitely many solutions.
3. Proportions
A proportion sets two fractions equal to each other. To solve, cross-multiply: if a/b = c/d, then a × d = b × c.
Worked Example: Solve 34 = x60.
Cross-multiply: 3 × 60 = 4 × x.
180 = 4x.
Divide by 4: x = 45
1. Solve 34 = x60. (see worked example above)
Explanation: Cross-multiply: 3 × 60 = 4 × x → 180 = 4x → x = 45.
2. Solve 58 = x40.
Explanation: Cross-multiply: 5 × 40 = 8 × x → 200 = 8x → x = 25.
3. Solve x9 = 1227.
Explanation: Cross-multiply: x × 27 = 9 × 12 → 27x = 108 → x = 4.
4. Word Problems with Linear Models
Many PSAT 8/9 word problems describe a real situation with a starting value and a constant rate of change. Translate the situation into an equation first (often total = rate × quantity + starting value, or profit = revenue − expenses), then solve.
Worked Example: A rental company charges \$14 per tablet. After renting out some tablets and paying \$112 in expenses, the company's total profit is \$406. How many tablets were rented?
Let n = number of tablets. Revenue = 14n. Profit = Revenue − Expenses: 14n − 112 = 406.
Add 112: 14n = 518.
Divide by 14: n = 37 tablets
1. A rental company charges \$14 per tablet. After renting out some tablets and paying \$112 in expenses, the company's total profit is \$406. How many tablets were rented? (see worked example above)
Explanation: Revenue minus expenses equals profit: 14n − 112 = 406. Adding 112 gives 14n = 518, so n = 37.
2. A club starts with 75 members and gains 8 new members per day. How many members will the club have after 5 days?
Explanation: Members = starting value + rate × days = 75 + 8(5) = 75 + 40 = 115.
3. Solve 3(2x − 5) + x = 35 for x, then find the value of 2x.
Explanation: Distribute: 6x − 15 + x = 35. Combine like terms: 7x − 15 = 35. Add 15: 7x = 50. Divide by 7: x = 50/7 ≈ 7.14, so 2x ≈ 14.29.
Common Mistakes to Avoid
- Forgetting to distribute to every term inside a parenthesis, especially when the term outside is negative, which flips the sign of every term inside.
- Stopping too early when the variable terms cancel out. If the remaining statement is false (like −22 = 1), the equation has no solution; if it's always true (like 6 = 6), it has infinitely many solutions.
- Cross-multiplying incorrectly in a proportion by multiplying numerator by numerator instead of numerator by the opposite denominator.
- Answering with the wrong variable or expression. Some questions ask you to solve for x but then report a different expression, like 2x or x + 3, always check exactly what's being asked before selecting an answer.
- Misreading a word problem's starting value versus its rate, such as confusing a one-time fee with a per-unit charge.
Frequently Asked Questions
How do I know if a linear equation has no solution or infinitely many solutions?
Fully simplify both sides first. If the variable terms cancel out and you're left with a true statement (like 5 = 5), the equation has infinitely many solutions. If you're left with a false statement (like 3 = 7), it has no solution.
What's the fastest way to solve a proportion?
Cross-multiplying is usually fastest: for a/b = c/d, multiply a by d and b by c, set those products equal, and solve the resulting equation.
Where can I practice more problems like these?
The Linear Equations quizzes in the PSAT 8/9 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 8/9. You can also browse the full PSAT 8/9 Math Question Bank.