Question

from unit 3, lesson 12
the points with coordinates (4, 8), (2, 10), and (5, 7) all lie on the line 2x + 2y = 24.

a. create a graph, plot the points, and sketch the line.

b. what is the slope of the line you graphed?

c. what does this slope tell you about the relationship between lengths and widths of rectangles with perimeter 24 units?

Explanation:

Part b

Step1: Rewrite the line equation in slope - intercept form ($y = mx + b$, where $m$ is the slope and $b$ is the y - intercept)

We start with the equation $2x + 2y=24$. First, we divide each term in the equation by 2 to simplify it: $\frac{2x}{2}+\frac{2y}{2}=\frac{24}{2}$, which gives us $x + y = 12$. Then, we solve for $y$ by subtracting $x$ from both sides: $y=-x + 12$.

Step2: Identify the slope

In the slope - intercept form $y = mx + b$, the coefficient of $x$ is the slope. For the equation $y=-x + 12$, the coefficient of $x$ is $- 1$. So the slope $m=-1$.

The equation of the line $2x + 2y = 24$ can be related to the perimeter formula of a rectangle. The perimeter of a rectangle is $P = 2l+2w$ (where $l$ is the length and $w$ is the width). If we let $x$ represent the length and $y$ represent the width, then the equation $2x + 2y=24$ is the perimeter formula for a rectangle with a perimeter of 24 units. The slope of the line $y=-x + 12$ is - 1. A slope of - 1 means that for every 1 - unit increase in the length ($x$), the width ($y$) decreases by 1 unit. In other words, the length and the width of a rectangle with a perimeter of 24 units are negatively related, and their rate of change with respect to each other is - 1.

Answer:

The slope of the line is \(-1\).

Part c