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Linear equations in one variable

Solve one-variable linear equations, including ones with fractions, parentheses, or no solution at all.

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Practice linear equations in one variable free

What the SAT actually asks

You get a single equation with one unknown and have to solve it or describe its solutions. The equation may have parentheses to distribute, fractions or decimals to clear, variables on both sides, or an unknown constant that controls how many solutions exist. Some questions are pure algebra. Others wrap the equation in a short real-world setup and ask for the value of the unknown, or for a quantity built from it. A few ask when an equation has no solution or infinitely many solutions.

Algebra is about 35 percent of the 44 math questions, and one-variable linear equations usually account for roughly 2 to 4 of them, though the exact count moves from test to test.

Where students lose points

How to attack these questions

  1. 1Clear fractions first by multiplying every term on both sides by the common denominator, so you never carry fractions through the whole problem.
  2. 2Collect all variable terms on one side and all constants on the other in one deliberate move rather than several small ones.
  3. 3Reread the last line of the question before you bubble, and if it asks for an expression, plug your value in and compute it.

Example questions

Written for this skill, with the answer explained. 89 more are in the app.

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Linear equations in one variable
Real question

If 2/3 x = 8, what is x?

Easier

The equation 3(x + 1) = 3x + 5 has how many solutions?

  • A) 1
  • B) 0 ✓
  • C) Infinite
  • D) 2

Why: Distribute the 3 on the left: 3x + 3 = 3x + 5. Subtract 3x from both sides: 3 = 5, which is false, so no value of x works and the number of solutions is 0. Answering 1 fits an equation whose x-terms differ, such as 4x + 3 = 3x + 5, but here the x-terms cancel. Answering Infinite would require the constants to match as well, as in 3(x + 1) = 3x + 3, and a linear equation can never have exactly 2 solutions.

Easier

The sum of three consecutive integers is 45. The largest is:

  • A) 14
  • B) 15
  • C) 16 ✓
  • D) 17

Why: Call the integers n, n + 1 and n + 2. Their sum is 3n + 3 = 45, so 3n = 42 and n = 14. The three integers are 14, 15 and 16, so the largest is 16. Answering 14 gives the smallest of the three and 15 gives the middle one, which is also 45 divided by 3. Answering 17 comes from starting the run at 15, and 15 + 16 + 17 = 48, which is too large.

Harder

If 5(x - 2) + 3 = 2(x + 4) - 6, what is the value of 4x?

  • A) 3
  • B) 9
  • C) 12 ✓
  • D) 20

Why: Expand both sides: 5x - 7 = 2x + 2. Then 3x = 9 and x = 3, so 4x = 12. Choice B is the 3x value you pass through on the way, and choice A is x itself. Answering 3 means you solved for x but the question asked for 4x. Answering 9 means you reported the value of 3x from the next-to-last step. Answering 20 means you treated the +3 on the left as -3, getting 5x - 13 = 2x + 2.

Questions students ask

How do I know if a linear equation has no solution?

Simplify both sides completely. If the variable terms cancel and you are left with a false numeric statement such as 3 equals 8, there is no solution. If the variables cancel and you are left with a true statement such as 5 equals 5, every real number is a solution.

Can I just plug the answer choices into the equation?

Yes, and it is often faster when the algebra looks messy. Start with the middle choices so you can rule out a direction quickly. Backsolving does not work on grid-in questions, so you still need the algebra for those.

What does it mean when an equation has an unknown constant like k?

The question is usually asking you to control the number of solutions. Simplify until you have something of the form ax equals b, then set the coefficient a to zero for the no-solution or infinite-solution case and solve for the constant.

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Questions on linear equations in one variable alone, with explanations on every one, and the app tracks which ones you keep missing.

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