Concept 5

Build a system from a context when two relationships are described.

Use perimeter and area to build a two-equation system for a rectangle.

Core Idea

Two unknowns + two relationships = two equations. Translate each sentence describing a relationship into its own equation, then solve the system.

Understanding

Word problems on the SAT describe two quantities connected by two different relationships. Rule: assign variables, write one equation per relationship, then solve.

  1. Read the problem once to identify the two unknowns.
  2. Read again to turn each relationship into an equation.
  3. Use substitution, because the SAT usually gives one simple relationship and one more complicated one.
  4. Check the answer in context: lengths must be positive, counts must be whole numbers.

Common setups include sum and product, perimeter and area, and cost and quantity.

Step by Step

  1. Read the problem and identify the two unknown quantities. Assign variables.
  2. Find the first relationship and write it as an equation.
  3. Find the second relationship and write it as a separate equation.
  4. Solve the system using substitution.
  5. Check that your answer makes sense in context (positive lengths, whole counts, etc.).

Misconceptions

  • Writing only one equation and trying to solve with one variable — two unknowns require two equations.
  • Mixing up which quantities go into which relationship (e.g., swapping length and width in a perimeter equation).
  • Keeping a negative solution when the context requires a positive value.
Question

Worked Example

A rectangle has a perimeter of 28 centimeters and an area of 48 square centimeters. What is the length, in centimeters, of the longer side of the rectangle?

Select an answer to see the explanation