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 bike-share program charges $15 to unlock a bike plus $5 per mile ridden. If Lena paid a total of $70, how many miles were paid for?
- A.
  11
- B.
  12
- C.
  14
- D.
  17
Answer: A. 11
Let n be the number of miles. Then 15 + 5n = 70. Subtract 15: 5n = 55. Divide by 5: n = 11.
Question 3
Each month Rahim spends 1/3 of his allowance on transport and 1/4 of his allowance on lunches. Together, these two expenses total 14 dollars each month. What is Rahim's monthly allowance, in dollars?
- A.
  2
- B.
  8
- C.
  24
- D.
  98
Answer: C. 24
Let the allowance be A. Then (1/3)A + (1/4)A = 14, so (7/12)A = 14 and A = 14 × 12/7 = 24. Check: 24/3 + 24/4 = 8 + 6 = 14. D adds the denominators to get 1/7 and solves A/7 = 14; B multiplies 14 by 7/12 instead of dividing; A divides 14 by 7 as if the two fractions summed to 7.
Question 4
In the equation (k − 2)x + 9 = 3x + 4k − 11, k is a constant. A perfume maker uses this equation to balance two models of how much fragrance remains in a bottle. For which value of k does the equation have infinitely many solutions?
- A.
  1
- B.
  3
- C.
  5
- D.
  7
Answer: C. 5
Infinitely many solutions need identical sides: the x-coefficients give k − 2 = 3, so k = 5, and the constants give 9 = 4(5) − 11 = 9, which checks. A subtracts 2 the wrong way (3 − 2); B sets k equal to the coefficient 3 and ignores the −2; D adds 2 a second time to the correct 5. So the key is C.
Question 5
d = w²/(16f) For a solar cooker's parabolic mirror, the depth d of the mirror is related to its width w and its focal length f by the formula above. Which equation gives f in terms of d and w?
- A.
  f = 16d/w²
- B.
  f = w²/(16d)
- C.
  f = 16w²/d
- D.
  f = w²d/16
Answer: B. f = w²/(16d)
Multiply both sides by 16f: 16fd = w². Then divide by 16d: f = w²/(16d) (B). A inverts the whole fraction; C multiplies by 16 instead of dividing, putting 16 in the numerator; D multiplies by d instead of dividing by it.

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