Question

find an equation for the line that passes through the points (-1, 4) and (5, 6).

Explanation:

Step1: Calculate the slope

The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Given points \( (-1, 4) \) and \( (5, 6) \), we substitute \( x_1=-1,y_1 = 4,x_2 = 5,y_2=6 \) into the formula:
\( m=\frac{6 - 4}{5-(-1)}=\frac{2}{6}=\frac{1}{3} \)

Step2: Use point - slope form to find the equation

The point - slope form of a line is \( y - y_1=m(x - x_1) \). We can use the point \( (-1,4) \) and \( m=\frac{1}{3} \). Substitute these values into the formula:
\( y - 4=\frac{1}{3}(x-(-1)) \)
Simplify the equation:
\( y - 4=\frac{1}{3}(x + 1) \)
Expand the right - hand side: \( y-4=\frac{1}{3}x+\frac{1}{3} \)
Add 4 to both sides: \( y=\frac{1}{3}x+\frac{1}{3}+4 \)
Since \( 4=\frac{12}{3} \), then \( y=\frac{1}{3}x+\frac{1 + 12}{3}=\frac{1}{3}x+\frac{13}{3} \)
We can also write it in standard form \( Ax+By = C \). Multiply through by 3 to get \( 3y=x + 13 \), or \( x-3y=-13 \)

Answer:

The equation of the line is \( y=\frac{1}{3}x+\frac{13}{3} \) (or \( x - 3y=-13 \))