Solving linear equations in one variable is the most fundamental algebra skill tested on Digital SAT Math, but the exam rarely tests it in a purely mechanical way. Instead, it wraps the skill inside word problems, asks about the number of solutions a strange-looking equation has, tests absolute value equations that produce two possible answers, and asks you to solve for a compound expression like x + 9 or 4x − 4 rather than x itself — a shortcut that saves real time once you recognize it.
The 21 problems below cover every version of this skill that shows up on real, recently administered Digital SAT exams, including the classic no-solution and infinite-solutions traps, absolute value equations, and rate-based word problems. Every problem includes a complete step-by-step explanation, hidden by default so you can test yourself honestly before checking your work. Pay special attention to the problems that ask "how many solutions" rather than for a specific value — recognizing when both sides of an equation collapse to the same expression (infinite solutions) versus a false statement (no solution) is one of the most commonly missed distinctions on test day. For unlimited additional practice, our free SAT Math QBank has over 2,500 more problems across every SAT Math topic, each with instant explanations.
If (x + 9) − 5 = 25, what is the value of x + 9?
Add 5 to both sides to isolate x + 9 directly, without solving for x itself: (x + 9) − 5 + 5 = 25 + 5.
That gives x + 9 = 30. The correct answer is A.
Which equation has no solution?
Check each equation for whether a value of x can make it true. In option A, subtracting x from both sides gives 0 = 1, which is never true — no value of x works.
Options B and D each solve to a single value of x, and option C is true for every x (infinitely many solutions). Only option A has no solution. The correct answer is A.
|15 − x| − 9 = 0. What are all positive solutions to the given equation?
Isolate the absolute value: |15 − x| = 9. This splits into two cases: 15 − x = 9 or 15 − x = −9.
The first case gives x = 6, and the second gives x = 24. Both are positive. The correct answer is D.
ax − y = bxy. The given equation relates the positive numbers a, b, x, and y. Which equation correctly expresses a in terms of b, x, and y?
Isolate the ax term by adding y to both sides: ax = bxy + y.
Divide both sides by x: a = (bxy + y)/x. The correct answer is A.
What value of x is the solution to the equation x + 10 = 3?
Subtract 10 from both sides: x = 3 − 10.
That gives x = −7. The correct answer is B.
If 5(x + 1) = 25, what is the value of x + 1?
Divide both sides by 5 to isolate (x + 1) directly — there's no need to solve for x itself.
That gives x + 1 = 5. The correct answer is B.
|3x − 12| = 18. What is the sum of the solutions to the given equation?
Split into two cases: 3x − 12 = 18, giving 3x = 30 and x = 10; or 3x − 12 = −18, giving 3x = −6 and x = −2.
The sum of the two solutions is 10 + (−2) = 8. The correct answer is B.
(1/3)(6y − 9) = 13. What value of y satisfies the given equation?
Multiply both sides by 3 to clear the fraction: 6y − 9 = 39.
Add 9 to both sides: 6y = 48, so y = 8.
The cost to rent a bus from Company X is $950 for the first 3 hours and an additional $50 per hour for each hour after the first 3 hours. If the total cost to rent the bus from Company X for t hours is $1,250, which equation represents this situation?
The flat $950 covers the first 3 hours. Every additional hour beyond that first 3 costs $50, and the number of additional hours is (t − 3).
Total cost = 950 + 50(t − 3), set equal to 1,250. The correct answer is A.
6(x − 9) = 11(x − 9). What value of x is the solution to the given equation?
Notice both sides share the factor (x − 9). If x − 9 were nonzero, dividing both sides by it would give 6 = 11, which is false — so that case is impossible.
The only way the equation can be true is if x − 9 = 0, making both sides equal to 0. So x = 9.
How many solutions does the equation 10x = 10 have?
Divide both sides by 10: x = 1. This is a single specific value that satisfies the equation.
A basic linear equation like this one has exactly one solution. The correct answer is B.
|6x − 3| = |5x|. What is the smallest solution to the given equation?
When two absolute values are equal, the expressions inside are either equal or opposites. Case 1: 6x − 3 = 5x, giving x = 3. Case 2: 6x − 3 = −5x, giving 11x = 3, so x = 3/11.
Comparing 3 and 3/11, the smaller value is x = 3/11.
9x + x = 2x + x + 5. Which of the following equations has the same solution as the given equation?
Simplify both sides of the original equation first: the left side is 10x, and the right side is 3x + 5. So 10x = 3x + 5.
Subtract 3x from both sides: 7x = 5. This matches option A exactly, meaning it has the same solution. The correct answer is A.
12x = 65 − x. What value of x is the solution to the given equation?
Add x to both sides to collect the x-terms: 12x + x = 65, so 13x = 65.
Divide both sides by 13: x = 5.
Gentoo penguins are the fastest swimming underwater birds. A gentoo penguin was recorded swimming at a rate of 6 meters per second for 110 meters. For approximately how many seconds was the penguin recorded swimming at this rate?
Time equals distance divided by rate: 110 meters ÷ 6 meters per second.
110/6 ≈ 18.3 seconds, which rounds to approximately 18 seconds. The correct answer is A.
If 2x + 3 = 9, what is the value of 4x − 4?
Solve for x first: 2x = 6, so x = 3.
Substitute into the target expression: 4(3) − 4 = 12 − 4 = 8.
If 2x + 3 = 9, what is the value of 4x − 4?
Solve for x first: 2x = 6, so x = 3.
Substitute into the target expression: 4(3) − 4 = 12 − 4 = 8.
If −9(15 − 3x) + 5 = −10(15 − 3x) + 21, what is the value of 15 − 3x?
Let u = 15 − 3x to simplify. The equation becomes −9u + 5 = −10u + 21.
Add 10u to both sides: u + 5 = 21, so u = 16. That means 15 − 3x = 16.
−5x + 15rx = 135. In the given equation, r is a constant. The equation has no solution. What is the value of r?
Factor out x on the left side: x(−5 + 15r) = 135. For this equation to have no solution, the coefficient of x must equal 0 — otherwise you could always divide both sides by it to find a value of x.
Set −5 + 15r = 0: 15r = 5, so r = 5/15 = 1/3. The correct answer is B.
A veterinarian recommends that each day a certain cat should eat 24 calories per pound of the cat's weight, plus an additional 14 calories. Which equation represents this situation, where c is the total number of calories the veterinarian recommends the cat should eat each day if the cat's weight is x pounds?
The calories scale with weight at a rate of 24 calories per pound, giving the term 24x. The extra 14 calories are a flat amount added on top, independent of weight.
Combining: c = 24x + 14. The correct answer is D.
How many solutions does the equation 9x + 6 = 5x + 4x + 12 have?
Simplify the right side: 5x + 4x + 12 = 9x + 12. The equation becomes 9x + 6 = 9x + 12.
Subtract 9x from both sides: 6 = 12, which is never true, regardless of x. Since no value of x can satisfy the equation, it has zero solutions. The correct answer is D.
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