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SAT systems of equations: satisfy both equations

Use substitution to solve a system, check both equations, and identify the quantity the question asks for.

SAT RISE · AI-assisted guide · Updated

The idea

A solution of a system of two linear equations in two variables is an ordered pair that satisfies both equations at the same time. Substitution replaces a variable with an equal expression from the other equation.

A step-by-step method

  1. For a word problem, name the unknown quantities and translate each stated total into an equation.
  2. Identify an equation that already isolates a variable, or isolate one using equivalent operations.
  3. Substitute that expression into the other equation and solve the resulting equation.
  4. Find the other variable and check the pair in both original equations. Answer the particular quantity requested.
  5. For a sum, difference, or other expression, calculate it from the checked pair rather than reporting only x or y.

Worked example

Attempt this original question, then open the reasoning to compare your approach.

The system is x + y = 10 and y = 2x + 1. What is x?

  1. 2
  2. 3
  3. 7
  4. 10
Show answer and reasoning

Answer: B · 3

Substitute y = 2x + 1 into x + y = 10: 3x + 1 = 10, so x = 3.

Then y = 7. Check: 3 + 7 = 10 and 7 = 2(3) + 1. The question asks for x, so the answer is 3.

Why each option works or fails

  1. x = 2 gives y = 5 from the second equation, but their sum is 7.
  2. Correct: the pair (3, 7) satisfies both equations.
  3. 7 is y, not the requested x.
  4. 10 is the sum x + y, not x.

Try a new question

Work independently before opening the explanation. Check what changed and whether the same rule still applies.

At a school performance, adult tickets cost $12 and student tickets cost $8. A total of 140 tickets were sold for $1,360. How many more student tickets than adult tickets were sold?

  1. 20
  2. 60
  3. 80
  4. 140
Show answer and reasoning

Answer: A · 20

Let a be adult tickets and s be student tickets. Then a + s = 140 and 12a + 8s = 1360.

Substitute s = 140 − a: 12a + 8(140 − a) = 1360, so 4a = 240. Thus a = 60 and s = 80.

The requested difference is s − a = 20. Check: 60 + 80 = 140 and 12(60) + 8(80) = 1360.

Why each option works or fails

  1. Correct: 80 student tickets minus 60 adult tickets gives 20.
  2. 60 is the adult ticket count, not the difference.
  3. 80 is the student ticket count, not the difference.
  4. 140 is the total number of tickets.

Common mistakes

  • Satisfying just one equation is insufficient.
  • A system may have one solution, no solution, or infinitely many solutions; substitution can reveal a contradiction or an identity.

Sources and related guides

This guide was prepared with AI assistance. Its rules and worked examples were checked against the references below; no independent expert review is claimed. Examples are original and are not official SAT questions.

Revision: Clarified checking both original equations before calculating the requested quantity.

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