Chapter 8.6, Problem 5E

To determine

To write:

Anequation of the line that passes through the given points.

Answer to Problem 5E

The equation of the line passing through the given pointswould be $y=\frac{2}{3}x-5$ .

Explanation of Solution

Given:

Two points on a line: $(0,-5)\text{and }(3,-3)$

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 $(0,-5)=(x_1,y_1)\text{and }(3,-3)=(x_2,y_2)$ . Upon substituting our given coordinates in slope formula, we will get:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{-3-(-5)}{3-0}\\ m=\frac{-3+5}{3}\\ m=\frac{2}{3}\end{array}$

To write our equation, we will substitute the value of slope and coordinates of point $(0,-5)$ in point-slope form of equation as shown below:

  $\begin{array}{l}y-y_1=m\left(x-x_1\right)\left(\text{Point-slope form of equation}\right)\\ y-(-5)=\frac{2}{3}\left(x-0\right)\left(\text{Substituting given values}\right)\\ y+5=\frac{2}{3}x\\ y+5-5=\frac{2}{3}x-5\\ y=\frac{2}{3}x-5\end{array}$

Therefore, the equation of the line passing through the given points would be $y=\frac{2}{3}x-5$ .