Solve systems of two linear equations using elimination or substitution, and reason about when systems have one, none, or infinite solutions.
83 original systems of linear equations questions in the SAT Ranker bank, with instant scoring and an Elo-based rank once you sign up. Below are a few free samples with full explanations.
At a snack stand, 3 pretzels and 2 lemonades cost $16, while 5 pretzels and 2 lemonades cost $22. What is the price, in dollars, of one pretzel?
Subtracting the first equation from the second eliminates the lemonades: (5p + 2l) − (3p + 2l) = 22 − 16, so 2p = 6 and p = 3. (A lemonade then costs $3.50, but the question asks for the pretzel.)
A system has infinitely many solutions: 4x + 6y = 12 and 2x + ky = 6. What is the value of k?
Divide the first equation by 2: 2x + 3y = 6. For infinitely many solutions, this must match the second equation exactly, so k = 3.
If 3x − 2y = 4 and x + 2y = 12, what is the value of x?
Adding the two equations eliminates y: (3x − 2y) + (x + 2y) = 4 + 12, so 4x = 16, giving x = 4.
If 3x + 2y = 11 and x + 4y = -23, what is the value of x?
Eliminating gives x = 9 and y = -8. The answer is 9. -8 solves the system correctly but reports y instead of x; -9 loses a sign while eliminating; 1 adds the two unknowns instead of reporting the one asked for.
If 6x + y = 48 and 2x + y = 20, what is the value of y?
Eliminating gives x = 7 and y = 6. The answer is 6. 7 solves the system correctly but reports x instead of y; -6 loses a sign while eliminating; 13 adds the two unknowns instead of reporting the one asked for.