Question

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

Explanation:

Step1: Calculate the slope

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \((x_1, y_1) = (2, -4)\) and \((x_2, y_2) = (-4, -2)\). So, \( m = \frac{-2 - (-4)}{-4 - 2} = \frac{2}{-6} = -\frac{1}{3} \).

Step2: Use point - slope form

The point - slope form of a line is \( y - y_1 = m(x - x_1) \). Let's use the point \((2, -4)\). Substituting \( m = -\frac{1}{3} \), \( x_1 = 2 \), and \( y_1 = -4 \) into the formula, we get \( y - (-4)=-\frac{1}{3}(x - 2) \).

Step3: Simplify the equation

Simplify \( y + 4 = -\frac{1}{3}x+\frac{2}{3} \). Subtract 4 from both sides: \( y=-\frac{1}{3}x+\frac{2}{3}-4 \). Since \( 4=\frac{12}{3} \), \( y = -\frac{1}{3}x+\frac{2 - 12}{3}=-\frac{1}{3}x-\frac{10}{3} \). We can also write it in standard form or other forms, but the slope - intercept form is \( y=-\frac{1}{3}x-\frac{10}{3} \).

Answer:

\( y = -\frac{1}{3}x-\frac{10}{3} \) (or equivalent forms like \( x + 3y=- 10 \))