Question

find an equation for the line that passes through the points (-4, -3) and (2, -1).

Explanation:

Step1: Calculate the slope (m)

The formula for slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1, y_1)=(-4, -3)\) and \((x_2, y_2)=(2, -1)\). Then \(m=\frac{-1 - (-3)}{2 - (-4)}=\frac{-1 + 3}{2 + 4}=\frac{2}{6}=\frac{1}{3}\).

Step2: 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 \((2, -1)\) (we could also use \((-4, -3)\)). Substitute \(m = \frac{1}{3}\), \(x_1 = 2\), and \(y_1=-1\) into the formula:
\(y-(-1)=\frac{1}{3}(x - 2)\)
Simplify the left - hand side: \(y + 1=\frac{1}{3}(x - 2)\)
Distribute the \(\frac{1}{3}\) on the right - hand side: \(y+1=\frac{1}{3}x-\frac{2}{3}\)
Subtract 1 from both sides: \(y=\frac{1}{3}x-\frac{2}{3}-1\)
Since \(1=\frac{3}{3}\), then \(y=\frac{1}{3}x-\frac{2}{3}-\frac{3}{3}=\frac{1}{3}x-\frac{5}{3}\)
We can also write it in standard form \(Ax+By = C\). Multiply through by 3 to get \(3y=x - 5\), or \(x-3y = 5\). But the slope - intercept form \(y=\frac{1}{3}x-\frac{5}{3}\) is also correct.

Answer:

The equation of the line is \(y=\frac{1}{3}x-\frac{5}{3}\) (or \(x - 3y=5\))