Algebra · Math

Systems of two linear equations

Systems of two linear equations: solving by substitution or elimination, interpreting a solution in context, and finding a parameter that makes a system have no solution or infinitely many.

30 questions in the bank · 7 easier · 15 medium · 8 harder · this domain is ≈35% of the section

Learn it in the Guide

What to watch for

Check what the question asks for. Many of these solve cleanly to x and y and then ask for x + y, or for the other variable — stopping at the first value you find is the classic miss.

How the question is worded

These are the actual stems used in the bank — the SAT reuses a small set of wordings, and recognising them saves time on test day.

  • The solution to the given system of equations is (x, y). What is the value of x + y?
  • A total of 37 tickets to a school play were sold for $435. Each of the adult tickets cost $15 and each of the student tickets cost $7. How many of the adult tickets were sold?
  • A total of 28 boxes packed for a food drive were sold for $282. Each of the large boxes cost $12 and each of the small boxes cost $6. How many of the large boxes were sold?
  • A total of 56 passes sold for a robotics expo were sold for $328. Each of the premium passes cost $8 and each of the standard passes cost $4. How many of the premium passes were sold?

Worked examples

easyAccording to the given system of equations, what is the value of x?

According to the given system of equations, what is the value of x?

  • A. 2
  • B. 3
  • C. 5
  • D. 12

Answer: B. 3

Explanation

Adding the two equations eliminates y: (x + y) + (3x − y) = 5 + 7, so 4x = 12 and x = 3.

mediumThe solution to the given system of equations is (x, y). What is the value of …

The solution to the given system of equations is (x, y). What is the value of x + y?

  • A. −1
  • B. 4
  • C. 5
  • D. 9

Answer: D. 9

Explanation

Eliminate one variable: multiplying and subtracting the equations gives x = 4 and y = 5. The question asks for x + y = 9 — the single most common miss here is stopping at x.

hardIn the given system of equations, k is a constant. If the system has no soluti…

In the given system of equations, k is a constant. If the system has no solution, what is the value of k?

Answer: 8

Explanation

Multiply the first equation by 2: 8x + 6y = 38. The second equation has the same y-coefficient but a different constant (41 ≠ 38), so the lines must be parallel and distinct: matching x-coefficients forces k = 8.