Question

Complete the equation of the line through (2,1) and (5,-8) Use exact numbers

Answer 1

Given two points (x₁, y₁) and (x₂, y₂), the slope of the line that passes through these two points is given by the equation:

$m=\frac{y_2-y_1}{x_2-x_1}$

Now, for our problem, we identify these two points:

$\begin{gathered} (x_1,y_1)=(2,1) \\ (x_2,y_2)=(5,-8) \end{gathered}$

Using the previous equation to calculate the slope of the line:

$\begin{gathered} m=\frac{-8-1}{5-2}=\frac{-9}{3} \\ \Rightarrow m=-3 \end{gathered}$

Now, using the point-slope form of the line equation:

$(y-y_0)=m(x-x_0)$

Where (x₀, y₀) is any point of the line. Choosing (2, 1) as (x₀, y₀):

$\begin{gathered} (y-1)=-3(x-2) \\ y=-3x+6+1 \\ \therefore y=-3x+7 \end{gathered}$