Question

find an equation for the line below.

Explanation:

Step1: Identify two points on the line

From the graph, we can see two points: \((-5, -5)\) and \((-1, 1)\) (we can also use other points, but these are clear).

Step2: Calculate the slope \(m\)

The formula for slope is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1, y_1) = (-5, -5)\) and \((x_2, y_2) = (-1, 1)\). Then:

$$ m=\frac{1 - (-5)}{-1 - (-5)}=\frac{1 + 5}{-1 + 5}=\frac{6}{4}=\frac{3}{2} $$

Step3: Use point - slope form to find the equation

We can use the point - slope form \(y - y_1=m(x - x_1)\). Let's use the point \((-1, 1)\) and \(m = \frac{3}{2}\).

$$ y - 1=\frac{3}{2}(x - (-1)) $$

$$ y - 1=\frac{3}{2}(x + 1) $$

Expand the right - hand side:

$$ y - 1=\frac{3}{2}x+\frac{3}{2} $$

Add 1 to both sides (note that \(1=\frac{2}{2}\)):

$$ y=\frac{3}{2}x+\frac{3}{2}+\frac{2}{2} $$

$$ y=\frac{3}{2}x+\frac{5}{2} $$

We can also verify with the y - intercept. From the graph, when \(x = 0\), \(y=\frac{5}{2}=2.5\), which matches the equation.

Answer:

The equation of the line is \(y=\frac{3}{2}x+\frac{5}{2}\) (or we can write it in standard form \(3x - 2y=- 5\), but the slope - intercept form is more common here)