Q 2. - Precalculus Enhanced with Graphing Utilities

Question

Find an equation of the line containing the points $(1, 4)$ and $(3, 8)$.

Step-by-Step Solution

Answer

The equation of the line is $\begin{array}{rcl}y& =& 2x+2\end{array}$.

Step 1. Given information

The line containing the points $(1, 4)$ and $(3, 8)$.

Step 2. Find the equation of the line that contain the given points.

The Point-Slope formula with point $(x_1, y_1)$ and the slope $m$ is given by $y-y_1=m(x-x_1)$, where $m$ is given by $m=\frac{y_2-y_1}{x_2-x_1}$.

Consider the two points as follows:

$\begin{array}{rcl}(x_1, y_1)& =& (1,4)\\ (x_2, y_2)& =& (3,8)\end{array}$

Find the slope.

$\begin{array}{rcl}m& =& \frac{y_2-y_1}{x_2-x_1}\\ & =& \frac{8-4}{3-1}\\ & =& \frac{4}{2}\\ & =& 2\end{array}$

Thus, the slope is $2$.

Step 3. Apply the Point-Slope formula to find the equation.

Substitute the value of the point $(x_1, y_1)$ and the slope into the Point-Slope formula.

$\begin{array}{rcl}y-4& =& 2(x-1)\\ y-4& =& 2x-2\\ y& =& 2x-2+4\\ y& =& 2x+2\end{array}$

Hence, the equation is $\begin{array}{rcl}y& =& 2x+2\end{array}$.