Math Problem Statement

The length of a rectangle is 4 times its width. The perimeter is 80. What system of equations can be solved to find the length and the width of the rectangle?

Solution

To find the system of equations that represents this problem, we need to break it down:

  1. Define the variables:

    • Let ww be the width of the rectangle.
    • Let ll be the length of the rectangle.

  2. Relationships given in the problem:

    • The length is 4 times the width: l=4wl = 4w.
    • The perimeter of a rectangle is 80, and the formula for the perimeter is: P=2l+2wP = 2l + 2w Substituting P=80P = 80, we have: 2l+2w=802l + 2w = 80

Thus, the system of equations to solve is:

1. \ & l = 4w \\ 2. \ & 2l + 2w = 80 \end{aligned}$$ This system can be solved to find the values of $$l$$ (length) and $$w$$ (width). Would you like me to solve this system and explain the steps? ### Related Questions: 1. What are the specific solutions for the length and width of this rectangle? 2. How would the system of equations change if the length were 3 times the width instead of 4? 3. Can the same approach be applied to find the dimensions of other shapes with known perimeters? 4. How can the solution be verified by substituting back into the original equations? 5. What would happen if the perimeter was given in terms of algebraic expressions rather than a constant value? **Tip**: Always start by defining variables and writing relationships explicitly; this simplifies creating the equations.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Geometry
Algebra

Formulas

Perimeter of a rectangle: P = 2l + 2w
Relationship: l = 4w

Theorems

Basic equation solving in algebra

Suitable Grade Level

Grades 7-8

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