Question

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

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 \((3, 1)\) and \( m = \frac{1}{2} \). Substituting these values, we get \( y - 1=\frac{1}{2}(x - 3) \).

Step3: Simplify to slope - intercept form

Expand the right - hand side: \( y - 1=\frac{1}{2}x-\frac{3}{2} \). Then, add 1 to both sides. Since \( 1=\frac{2}{2} \), we have \( y=\frac{1}{2}x-\frac{3}{2}+\frac{2}{2}=\frac{1}{2}x-\frac{1}{2} \). We can also write it in standard form or other forms, but the slope - intercept form is a common way. Another way is to use the two - point form or convert to standard form \( x - 2y = 1 \) (by multiplying both sides of \( y=\frac{1}{2}x-\frac{1}{2} \) by 2: \( 2y=x - 1 \), then \( x-2y = 1 \)). But the slope - intercept form \( y=\frac{1}{2}x-\frac{1}{2} \) is also correct.

Answer:

\( y=\frac{1}{2}x-\frac{1}{2} \) (or \( x - 2y=1 \) or other equivalent forms)