Question

find an equation for the line that passes through the points (5, -3) and (-1, -1).

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, \((x_1, y_1) = (5, -3)\) and \((x_2, y_2) = (-1, -1)\). So, \( m = \frac{-1 - (-3)}{-1 - 5} = \frac{-1 + 3}{-6} = \frac{2}{-6} = -\frac{1}{3} \).

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 \((5, -3)\). Substitute \( m = -\frac{1}{3} \), \( x_1 = 5 \) and \( y_1 = -3 \) into the formula:
\( y - (-3)=-\frac{1}{3}(x - 5) \)
Simplify the left - hand side: \( y + 3 = -\frac{1}{3}(x - 5) \)
Distribute the slope on the right - hand side: \( y+3 = -\frac{1}{3}x+\frac{5}{3} \)
Subtract 3 from both sides. Since \( 3=\frac{9}{3} \), we have \( y=-\frac{1}{3}x+\frac{5}{3}-\frac{9}{3} \)
Simplify the right - hand side: \( y = -\frac{1}{3}x-\frac{4}{3} \)
We can also write it in standard form \( x + 3y=-4 \) (by multiplying both sides by 3: \( 3y=-x - 4 \), then \( x+3y=-4 \))

Answer:

One possible equation is \( y = -\frac{1}{3}x-\frac{4}{3} \) (or in standard form \( x + 3y=-4 \))