Chapter 8.6, Problem 34E

To determine

To write:

An equation of the line that passes through the given points. Describe the method you used to determine the equation.

Answer to Problem 34E

The equation of the line passing through the given points would be $y+1=\frac{3}{2}\left(x-2\right)$ .

Explanation of Solution

Given:

Two points on a line: $(2,-1)\text{and }(6,5)$

Concept used:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1},\text{where,}\\ m=\text{Slope of line,}\\ y_2-y_1=\text{Difference between two }y\text{-coordinates}\\ x_2-x_1=\text{Difference between two }x\text{-coordinates of same }y-\text{coordinates}\end{array}$

The equation of a line in point-slope form is $y-y_1=m\left(x-x_1\right)$ , where m represents slope of line and $\left(x_1,y_1\right)$ represents a point on the line.

Calculation:

First of all, we will find the slope of line passing through the given points.

Let point $(2,-1)=(x_1,y_1)\text{and }(6,5)=(x_2,y_2)$ . Upon substituting our given coordinates in slope formula, we will get:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{5-(-1)}{6-2}\\ m=\frac{5+1}{4}\\ m=\frac{6}{4}\\ m=\frac{3}{2}\end{array}$

Now we will use point-slope form of equation of line to write our equation as we have figured out slope of line and we have been given a point on line.

Upon substituting the value of slope and coordinates of point $(2,-1)$ in point-slope form of equation, we will get:

  $\begin{array}{l}y-y_1=m\left(x-x_1\right)\left(\text{Point-slope form of equation}\right)\\ y-(-1)=\frac{3}{2}\left(x-2\right)\left(\text{Substituting given values}\right)\\ y+1=\frac{3}{2}\left(x-2\right)\end{array}$

Therefore, the equation of the line passing through the given points would be $y+1=\frac{3}{2}\left(x-2\right)$ .