Question

use the substitution method to determine the solution type of the following system.
3y - 6x = 3

y = 2x + 1

○ infinitely many solutions
○ (2, 0)
○ no solutions
○ (3, -1)

question 8 (5 points)
the perimeter of a rectangle field is 48 yards. the length is 6 yards more than the width. find the length and width of the rectangle field.

○ length = 15 yards; width = 9 yards
○ length = 9 yards; width = 3 yards
○ length = 11 yards; width = 5 yards
○ length = 10 yards; width = 4 yards

Explanation:

First Problem (System of Equations)

Step1: Substitute \( y = 2x + 1 \) into \( 3y - 6x = 3 \)

Substitute \( y \) in the first equation: \( 3(2x + 1) - 6x = 3 \)

Step2: Simplify the equation

Expand: \( 6x + 3 - 6x = 3 \)
Simplify: \( 3 = 3 \)
This is a true statement, meaning the two equations represent the same line, so there are infinitely many solutions.

Let the width be \( w \) yards, then the length is \( w + 6 \) yards. The perimeter of a rectangle is \( P = 2(l + w) \), and \( P = 48 \).

Step1: Set up the equation

\( 2((w + 6) + w) = 48 \)

Step2: Simplify the equation

Simplify inside the parentheses: \( 2(2w + 6) = 48 \)
Divide both sides by 2: \( 2w + 6 = 24 \)
Subtract 6: \( 2w = 18 \)
Divide by 2: \( w = 9 \)

Step3: Find the length

Length \( l = w + 6 = 9 + 6 = 15 \) yards.

Answer:

Infinitely many solutions

Second Problem (Rectangle Perimeter)