Question

find an equation for the line that passes through the points (3, -3) and (-5, -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, \((x_1, y_1) = (3, -3)\) and \((x_2, y_2) = (-5, -1)\). So, \( m = \frac{-1 - (-3)}{-5 - 3} = \frac{-1 + 3}{-8} = \frac{2}{-8} = -\frac{1}{4} \).

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 \((3, -3)\). Substitute \( m = -\frac{1}{4} \), \( x_1 = 3 \) and \( y_1 = -3 \) into the formula:
\( y - (-3)=-\frac{1}{4}(x - 3) \)
Simplify the left - hand side: \( y + 3 = -\frac{1}{4}(x - 3) \)
Expand the right - hand side: \( y+3 = -\frac{1}{4}x+\frac{3}{4} \)
Subtract 3 from both sides: \( y=-\frac{1}{4}x+\frac{3}{4}-3 \)
Since \( 3=\frac{12}{4} \), we have \( y = -\frac{1}{4}x+\frac{3 - 12}{4}=-\frac{1}{4}x-\frac{9}{4} \)
We can also write it in standard form \( x + 4y=-9 \) (by multiplying both sides of \( y = -\frac{1}{4}x-\frac{9}{4} \) by 4: \( 4y=-x - 9 \), then \( x + 4y=-9 \)) or in slope - intercept form \( y=-\frac{1}{4}x-\frac{9}{4} \)

Answer:

The equation of the line is \( y = -\frac{1}{4}x-\frac{9}{4} \) (or \( x + 4y=-9 \))