Question

find an equation for the line below.

Explanation:

Step1: Identify two points on the line

From the graph, we can see two points: let's take \((-3, 5)\) (the left green dot) and \((5, -1)\) (the right green dot).

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

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
Substituting the points: \(m = \frac{-1 - 5}{5 - (-3)} = \frac{-6}{8} = -\frac{3}{4}\).

Step3: Use point - slope form to find the equation

We use the point - slope form \(y - y_1 = m(x - x_1)\). Let's use the point \((-3, 5)\).
Substitute \(m = -\frac{3}{4}\), \(x_1=-3\), and \(y_1 = 5\) into the formula:
\(y - 5=-\frac{3}{4}(x + 3)\)
Expand the right - hand side: \(y - 5=-\frac{3}{4}x-\frac{9}{4}\)
Add 5 to both sides: \(y=-\frac{3}{4}x-\frac{9}{4}+5\)
Convert 5 to fourths: \(5=\frac{20}{4}\), so \(y = -\frac{3}{4}x+\frac{20 - 9}{4}=-\frac{3}{4}x+\frac{11}{4}\) (we can also verify with the other point. Let's check \((5,-1)\):
Left - hand side: \(y=-1\)
Right - hand side: \(-\frac{3}{4}(5)+\frac{11}{4}=-\frac{15}{4}+\frac{11}{4}=-\frac{4}{4}=-1\), which works. We can also write it in standard form or slope - intercept form. Another way is to use the slope - intercept form \(y=mx + b\) and solve for \(b\). We know \(m = -\frac{3}{4}\), and we can use one of the points, say when \(x = 0\), from the slope - intercept form, we can also calculate \(b\). Let's use the point \((-3,5)\) in \(y=mx + b\):
\(5=-\frac{3}{4}(-3)+b\)
\(5=\frac{9}{4}+b\)
\(b = 5-\frac{9}{4}=\frac{20 - 9}{4}=\frac{11}{4}\). So the equation is \(y=-\frac{3}{4}x+\frac{11}{4}\) or we can multiply through by 4 to get \(4y=-3x + 11\) or \(3x+4y = 11\). But the slope - intercept form is more common here.

Answer:

The equation of the line is \(y = -\frac{3}{4}x+\frac{11}{4}\) (or equivalent forms like \(3x + 4y=11\))