SAT · Math · Algebra

Linear equations in one variable

Linear equations in one variable is part of Algebra in the SAT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.

Question types you will see here include multi-step linear equation, no solution or infinitely many, rate plus fixed fee and solve for a related expression.

Common traps: sign error when distributing a negative; combining unlike terms; moving a term without changing its sign.

Sample questions

Question 1
A climbing gym charges a one-time membership fee of $15 plus $4 per visit. If Priya paid a total of $75, how many visits were paid for?
- A.
  15
- B.
  16
- C.
  18
- D.
  22
Answer: A. 15
Let n be the number of visits. Then 15 + 4n = 75. Subtract 15: 4n = 60. Divide by 4: n = 15.
Question 2
A running track is built from two straight sections and two semicircles. In the equation 2(x + 6) = 2x + k, x is the track's straight length in metres and k is a constant. For which value of k does the equation have infinitely many solutions?
- A.
  2
- B.
  6
- C.
  8
- D.
  12
Answer: D. 12
Distributing gives 2x + 12 = 2x + k. The x-terms already match, so the equation is true for all x exactly when k = 12 (D). A picks the multiplier 2; B picks the constant inside the parentheses; C adds 2 and 6.
Question 3
A concert's lighting rig is hung using the equation 3x − k = 2(x + 4), where k is a constant. If x = 15 is a solution, what is the value of k?
- A.
  −23
- B.
  −7
- C.
  7
- D.
  11
Answer: C. 7
Substitute x = 15: 45 − k = 2(19) = 38, so k = 7 (C). Choice A forgets to multiply x by 3 (15 − k = 38), B has a sign slip on the last step, and D distributes the 2 only to x (2x + 4 = 34, giving 45 − k = 34).
Question 4
A vegetable-oil recycling drive's total after r rounds satisfies 5(2r − 7) − 7 = 3r + 14. What is the value of r?
- A.
  4
- B.
  6
- C.
  8
- D.
  56
Answer: C. 8
Distribute: 10r − 35 − 7 = 3r + 14, so 7r = 56 and r = 8 (C). Choice A forgets to multiply the 7 by 5, giving 7r = 28; B turns −7 into +7 after distributing, giving 7r = 42; D stops at 7r = 56.
Question 5
(x + 4)/2 - (x - 1)/3 = 2 What is the solution to the equation above?
- A.
  -2
- B.
  2
- C.
  6
- D.
  26
Answer: A. -2
A is correct: Multiply by 6: 3(x + 4) - 2(x - 1) = 12, so x + 14 = 12 and x = -2. B does not distribute the minus sign over (x - 1). C multiplies 4 by 3 incorrectly, giving 3x + 4. D adds 14 to 12 instead of subtracting.

Find your real gaps

A few questions show the style. The free scan shows which SAT skills cost you the most points, then builds a plan around them.

Take the free scan