Question

find an equation for the line that passes through the points (-2, 4) and (6, -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 \((x_2,y_2)=(6,-2)\). So \( m=\frac{-2 - 4}{6 - (-2)}=\frac{-6}{8}=-\frac{3}{4} \).

Step2: Use point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \). Using the point \((-2,4)\) and \( m =-\frac{3}{4} \), we get \( y - 4=-\frac{3}{4}(x + 2) \).

Step3: Simplify to slope - intercept form

Expand the right - hand side: \( y-4=-\frac{3}{4}x-\frac{3}{2} \). Then add 4 to both sides. \( y=-\frac{3}{4}x-\frac{3}{2}+4 \). Since \( 4=\frac{8}{2} \), we have \( y=-\frac{3}{4}x+\frac{8 - 3}{2}=-\frac{3}{4}x+\frac{5}{2} \). We can also write it in standard form \( 3x + 4y=10 \) (by multiplying both sides of \( y=-\frac{3}{4}x+\frac{5}{2} \) by 4: \( 4y=-3x + 10 \), then \( 3x+4y = 10 \)).

Answer:

The equation of the line can be written in slope - intercept form as \( y=-\frac{3}{4}x+\frac{5}{2} \) or in standard form as \( 3x + 4y = 10 \) (or other equivalent forms).