Solving a Rectangle Perimeter Problem Using a System of Equations by Substitution
Apply the seven-step problem-solving strategy for systems of linear equations to a geometry word problem involving the perimeter of a rectangle, using substitution to solve the resulting system.
Problem: The perimeter of a rectangle is 88. The length is five more than twice the width. Find the length and the width.
- Read the problem and draw a rectangle labeled with width and length .
- Identify: The length and width of the rectangle.
- Name: Let = the length and = the width.
- Translate into a system of equations. The perimeter formula gives the first equation, and the relationship between the dimensions gives the second:
- Solve using substitution. Because the second equation is already solved for , substitute for in the first equation:
Distribute: . Combine like terms: . Subtract : . Divide by : .
Substitute into the second equation: .
- Check: Does a rectangle with length and width have perimeter ? ✓
- Answer: The length is and the width is .
This example shows how a geometry word problem naturally produces a system of two equations — one from a formula (the perimeter equation ) and one from a verbal relationship between the unknowns (). Because the relationship equation is already solved for , substitution begins immediately at Step 2 of the substitution method. Compare this with the single-variable approach to similar problems, where the length must first be expressed in terms of the width before substituting into the perimeter formula — the systems approach lets each unknown keep its own variable, which some learners find more intuitive to set up.
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