Question

find an equation for the line below.

Explanation:

Step1: Identify two points on the line

From the graph, we can see two points: \((-4, 6)\) and \((2, 2)\).

Step2: Calculate the slope (\(m\))

The formula for slope is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1,y_1)=(-4,6)\) and \((x_2,y_2)=(2,2)\). Then \(m=\frac{2 - 6}{2 - (-4)}=\frac{-4}{6}=-\frac{2}{3}\).

Step3: Use point - slope form to find the equation

The point - slope form of a line is \(y - y_1=m(x - x_1)\). We can use the point \((2,2)\) and \(m =-\frac{2}{3}\).
Substitute into the formula: \(y - 2=-\frac{2}{3}(x - 2)\).
Expand the right - hand side: \(y - 2=-\frac{2}{3}x+\frac{4}{3}\).
Add 2 to both sides: \(y=-\frac{2}{3}x+\frac{4}{3}+2\).
Since \(2=\frac{6}{3}\), then \(y=-\frac{2}{3}x+\frac{4 + 6}{3}=-\frac{2}{3}x+\frac{10}{3}\). We can also verify with the y - intercept. When \(x = 0\), from the graph, the line crosses the y - axis at \(y=\frac{10}{3}\approx3.33\), and our equation gives \(y=\frac{10}{3}\) when \(x = 0\). Another way is to use the slope - intercept form \(y=mx + b\). We know \(m=-\frac{2}{3}\), and we can use the point \((0,\frac{10}{3})\) (the y - intercept) directly.

Answer:

The equation of the line is \(y =-\frac{2}{3}x+\frac{10}{3}\)