Question

the perimeter of a rectangle is 38 units. the length is 13 units longer than the width. what are the dimensions of the rectangle?
a) 16 by 3
b) 18 by 5
c) 12 by 7
d) 15 by 4
question 6 (5 points)
solve the following system of equations using the elimination method.
$9x - 5y = -16$
$-3x + 7y = -16$
a) $(0,-5)$
b) $(-2,-3)$
c) $(-4,-4)$
d) $(4,4)$

Explanation:

First Question (Rectangle Dimensions)

Step1: Recall perimeter formula

The perimeter \( P \) of a rectangle is given by \( P = 2(l + w) \), where \( l \) is length and \( w \) is width. Let \( w = x \), then \( l = x + 13 \). Given \( P = 38 \), so \( 2((x + 13) + x) = 38 \).

Step2: Simplify the equation

Simplify: \( 2(2x + 13) = 38 \) → \( 4x + 26 = 38 \) → \( 4x = 12 \) → \( x = 3 \). Then \( l = 3 + 13 = 16 \). So dimensions are 16 by 3.

Step1: Multiply to eliminate \( x \)

The equations are \( 9x - 5y = -16 \) and \( -3x + 7y = -16 \). Multiply the second equation by 3: \( -9x + 21y = -48 \).

Step2: Add the equations

Add \( 9x - 5y = -16 \) and \( -9x + 21y = -48 \): \( (9x - 9x) + (-5y + 21y) = -16 - 48 \) → \( 16y = -64 \) → \( y = -4 \).

Step3: Substitute \( y \) to find \( x \)

Substitute \( y = -4 \) into \( -3x + 7y = -16 \): \( -3x + 7(-4) = -16 \) → \( -3x - 28 = -16 \) → \( -3x = 12 \) → \( x = -4 \). So solution is \( (-4, -4) \).

Answer:

A) 16 by 3

Second Question (System of Equations)