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 a

To create the graph:

1. Set up the coordinate system: Draw the x - axis (horizontal) and y - axis (vertical) with a suitable scale (e.g., 1 unit per grid square).
2. Plot the points:

• For the point \((4,8)\), move 4 units to the right along the x - axis and 8 units up along the y - axis and mark the point.
• For the point \((2,10)\), move 2 units to the right along the x - axis and 10 units up along the y - axis and mark the point.
• For the point \((5,7)\), move 5 units to the right along the x - axis and 7 units up along the y - axis and mark the point.

1. Sketch the line: The equation of the line is \(2x + 2y=24\). We can simplify it to \(y=-x + 12\) (by dividing both sides by 2 and then solving for \(y\): \(2y=-2x + 24\), \(y=-x + 12\)). To sketch the line, we can also find the x - intercept and y - intercept. When \(y = 0\), \(2x=24\), \(x = 12\) (x - intercept is \((12,0)\)). When \(x = 0\), \(2y=24\), \(y = 12\) (y - intercept is \((0,12)\)). Draw a straight line passing through the plotted points and the intercepts.

Part b

Step 1: Recall the slope formula

The slope \(m\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(m=\frac{y_2 - y_1}{x_2 - x_1}\).

Step 2: Choose two points

Let's choose the points \((4,8)\) and \((2,10)\). Here, \(x_1 = 4\), \(y_1 = 8\), \(x_2=2\), \(y_2 = 10\).

Step 3: Calculate the slope

Substitute the values into the slope formula: \(m=\frac{10 - 8}{2 - 4}=\frac{2}{-2}=- 1\). We can also use the simplified equation of the line \(y=-x + 12\). The slope - intercept form of a line is \(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. Comparing \(y=-x + 12\) with \(y=mx + b\), we see that \(m=-1\).

For a rectangle with perimeter \(P = 24\) units, the perimeter formula is \(P=2l + 2w\) (where \(l\) is the length and \(w\) is the width). The given equation \(2x+2y = 24\) can be related to the perimeter formula of a rectangle (if we let \(x\) represent the length and \(y\) represent the width or vice - versa). The slope of the line \(y=-x + 12\) is \(-1\). This slope tells us that for every 1 - unit increase in the length (x - value), the width (y - value) decreases by 1 unit, and vice - versa. In other words, the length and the width of a rectangle with a perimeter of 24 units are negatively related, and the rate of change of the width with respect to the length (or the length with respect to the width) is \(-1\).

Answer:

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

Part c