Question

challenge #4
plot four lines. each line should pass through the point on the ( y )-axis and one of the other four points. one is done for you.

line	equation
#2
#3
#4

(graph with grid, a red line, and several points)

Explanation:

Step1: Identify the y - intercept

The point on the y - axis (where \(x = 0\)) is \((0,-2)\), so the y - intercept \(b=-2\).

Step2: Choose another point

Let's take the point \((-4,3)\) (one of the blue points). The slope \(m\) between \((0, - 2)\) and \((-4,3)\) is calculated as \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{3-(-2)}{-4 - 0}=\frac{5}{-4}=-\frac{5}{4}\).

Step3: Write the equation

Using the slope - intercept form \(y=mx + b\), with \(m =-\frac{5}{4}\) and \(b=-2\), the equation is \(y=-2-\frac{5}{4}x\).

(Note: We can also choose other points. For example, if we take the point \((-4,-5)\), the slope \(m=\frac{-5 - (-2)}{-4-0}=\frac{-3}{-4}=\frac{3}{4}\), and the equation is \(y=-2+\frac{3}{4}x\). Or if we take the point \((2,5)\), the slope \(m=\frac{5-(-2)}{2 - 0}=\frac{7}{2}\), and the equation is \(y=-2+\frac{7}{2}x\). Or if we take the point \((0,5)\) (wait, no, the y - axis point is \((0,-2)\), another point: let's take \((-4,3)\) as above, or \((-4,-5)\), \((2,5)\), \((6,2)\) etc. Let's take \((-4,3)\) for line #2)

Answer:

\(y=-2-\frac{5}{4}x\) (or other valid equations based on different points, e.g., if we take \((-4,-5)\), \(y=-2+\frac{3}{4}x\); if we take \((2,5)\), \(y=-2+\frac{7}{2}x\); if we take \((6,2)\), \(m = \frac{2-(-2)}{6-0}=\frac{4}{6}=\frac{2}{3}\), equation \(y=-2+\frac{2}{3}x\))