Systems of two linear equations in two variables sample question 2

At 6 hours, both halls cost $112.

SAT · Math · Algebra

Systems of two linear equations in two variables

Systems of two linear equations in two variables 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 word problem with two unknowns, parallel versus infinite solutions, solve for a combination and meaning of the intersection point.

Common traps: swapping which variable is which; using one equation twice; solving for x when y is asked.

Sample questions

Question 1
-2x - 3y = -2 -2x - 4y = -2 If (x, y) is the solution to the system of equations above, what is the value of x + y?
- A.
  -1
- B.
  0
- C.
  1
- D.
  2
Answer: C. 1
Solve by elimination or substitution. The solution is x = 1, y = 0. Check: -2(1) + -3(0) = -2 and -2(1) + -4(0) = -2. Therefore x + y = 1.
Question 2
Hall A charges $40 plus $12 per hour. Hall B charges $70 plus $7 per hour. The graphs of the two cost functions intersect at the point (6, 112). What does this point mean?
- A.
  Hall A is cheaper for the first 6 hours.
- B.
  At 6 hours, both halls cost $112.
- C.
  Each hall charges $6 per hour.
- D.
  Hall B costs $112 more than Hall A.
Answer: B. At 6 hours, both halls cost $112.
B is correct: 40 + 12(6) = 112 and 70 + 7(6) = 112, so at 6 hours the two costs are equal at $112. A reads the intersection as a threshold only. C reads x as a rate. D reads y as a difference.
Question 3
A knot-tying rope is cut into two segments of lengths a and b metres, where a + b = 12 and a − b = 4. What is the value of a² − b²?
- A.
  16
- B.
  32
- C.
  48
- D.
  144
Answer: C. 48
Factor: a² − b² = (a + b)(a − b) = 12 × 4 = 48 (C). Choice A is 4², using only the difference; B is the product ab (a = 8, b = 4); D is 12², using only the sum.
Question 4
y = 2x − 3 5x − 3y = 4 A biscuit tin's stacking plan is described by the system above. If (x, y) is the solution of the system, what is the value of x?
- A.
  −7
- B.
  5
- C.
  7
- D.
  12
Answer: B. 5
Substitute: 5x − 3(2x − 3) = 4, so 5x − 6x + 9 = 4, −x = −5, and x = 5 (B). A distributes −3 to the 2x only, writing 5x − 6x − 3 = 4; C is the value of y; D is x + y, not x alone.
Question 5
3x + 5y = 23 5x + 3y = 17 The solution to the system of equations above is (x, y). What is the value of x − y?
- A.
  −3
- B.
  3
- C.
  5
- D.
  40
Answer: A. −3
Subtract the second equation from the first: −2x + 2y = 6, so y − x = 3 and x − y = −3 (the solution is x = 1, y = 4). B subtracts in the wrong order and reports y − x; C is x + y, found by adding and dividing by 8; D is the sum of the right-hand sides, 23 + 17.

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