Question

the perimeter of a rectangle is 32 meters and the length is 6 meters longer than the width. find the dimensions of the rectangle. let x = the length and y = the width. the corresponding modeling system is \\(\

$$\begin{cases} 2x + 2y = 32 \\\\ x - y = 6 \\end{cases}$$

\\). use the method of substitution to solve this system. answer length of the rectangle = \\(\square\\) m, width of the rectangle = \\(\square\\) m

Explanation:

Step1: Solve the second equation for x

From \( x - y = 6 \), we can express \( x \) in terms of \( y \) as \( x = y + 6 \).

Step2: Substitute x into the first equation

Substitute \( x = y + 6 \) into \( 2x + 2y = 32 \). We get \( 2(y + 6) + 2y = 32 \).

Step3: Simplify and solve for y

First, expand the left - hand side: \( 2y+12 + 2y=32 \). Combine like terms: \( 4y+12 = 32 \). Subtract 12 from both sides: \( 4y=32 - 12=20 \). Then divide both sides by 4: \( y=\frac{20}{4} = 5 \).

Step4: Find the value of x

Substitute \( y = 5 \) into \( x=y + 6 \), so \( x=5 + 6=11 \).

Answer:

Length of the rectangle = \( 11 \) m,
Width of the rectangle = \( 5 \) m