Linear equations on the SAT: lesson and practice
To solve a linear equation on the SAT, simplify each side, move the x-terms to one side and the numbers to the other, then divide by the coefficient of x. The SAT also asks when an equation has no solution (same x-coefficient, different constants) or infinitely many (both sides identical).
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How to solve a linear equation, step by step
- Clear brackets. Multiply the outside number by every term inside.
- Clear fractions by multiplying every term by the common denominator.
- Collect x-terms on one side and numbers on the other, doing the opposite operation each time something crosses the equals sign.
- Divide by the number in front of x.
- Check by putting your answer back into the original equation.
Example 1: brackets on one side
If 3(x − 4) = 2x + 5, what is x?
x − 12 = 5
x = 17
Check: 3(17 − 4) = 39 and 2(17) + 5 = 39. Both sides match.
Example 2: a word problem
A ride costs a booking fee of Rs 100 plus Rs 30 per kilometer. The total was Rs 400. How long was the ride?
30k = 300
k = 10 km
The fixed fee comes off before dividing. Dividing 400 by 30 is the most common trap.
No solution and infinitely many solutions
These are among the most-missed linear equation questions on the SAT. Simplify both sides fully, then compare.
| After simplifying | Example | Number of solutions |
|---|---|---|
| Different x-coefficients | 5x + 2 = 3x + 8 | Exactly one |
| Same x-coefficient, different constants | 6x + 8 = 6x + 5 | None |
| Both sides identical | 4x + 10 = 4x + 10 | Infinitely many |
Example 3: find k for no solution
In 2(3x + 4) = kx + 5, for what value of k is there no solution?
The constants (8 and 5) already differ, so we need equal x-coefficients: k = 6. Think of two parallel lines: same slope, never meeting.
The Desmos shortcut
The digital SAT has a built-in Desmos graphing calculator. Type the left side as one line and the right side as another. The x-value where they cross is the solution. Parallel lines mean no solution; one line on top of the other means infinitely many.
Common mistakes
The practice above tracks these for you. When you make one twice in a row, you get a targeted lesson and a new set of questions.
Sample questions with answers
1. If 3(x − 4) = 2x + 5, what is the value of x?
- 9
- 17 (correct)
- −7
- 1
Distribute to get 3x − 12 = 2x + 5, so x = 17. Choice A comes from writing 3x − 4.
2. For what value of k does 2(3x + 4) = kx + 5 have no solution?
- 3
- 6 (correct)
- 8
- −6
6x + 8 = kx + 5 has no solution when k = 6. Choice A forgets to distribute the 2.
3. A Rs 100 fee plus Rs 30 per km totals Rs 400. How many km?
- 13
- 10 (correct)
- 300
- 17
30k = 300, so k = 10. Choice A divides 400 by 30 and ignores the fee; choice C stops before dividing.
Frequently asked questions
How many linear equation questions are on the SAT?
Algebra makes up about 35% of SAT Math, roughly 13 to 15 of the 44 questions, and linear equations in one variable are one of its core question types. Many other questions also require solving a linear equation as a step.
When does a linear equation have no solution?
A linear equation has no solution when, after simplifying, both sides have the same x-coefficient but different constants, such as 6x + 8 = 6x + 5. It has infinitely many solutions when both sides are identical.
Can I use Desmos for linear equations on the SAT?
Yes. The digital SAT includes a built-in Desmos graphing calculator. Graph each side of the equation as its own line and read the x-value where they intersect. It is a fast way to check your algebra.
What is the most common mistake with linear equations on the SAT?
Not distributing to every term inside brackets, such as writing 3(x − 4) as 3x − 4 instead of 3x − 12, and sign errors when moving terms across the equals sign.