Chapter 4, Problem 13CR

To determine

To find: The $x$ -intercept of the line that passes through the points $\left(1,2\right)$ and $\left(2,7\right)$ .

Answer to Problem 13CR

The $x$ -intercept of the line that passes through the points $\left(1,2\right)$ and $\left(2,7\right)$ is $\frac{3}{5}$ that is Option C.

Explanation of Solution

Given:

The line passes through the points $\left(1,2\right)$ and $\left(2,7\right)$ .

Concept used:

The formula for the slope of the line $m$ containing points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is:

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

Calculation:

Replace $\left(x_1,y_1\right)$ as $\left(1,2\right)$ and $\left(x_2,y_2\right)$ as $\left(2,7\right)$ in the slope formula $m=\frac{y_2-y_1}{x_2-x_1}$ as:

$\begin{array}{c}m=\frac{7-2}{2-1}\\ =\frac{5}{1}\\ =5\end{array}$

The equation of the line with slope 5 and point $\left(1,2\right)$ is calculated as,

$\begin{array}{c}y-2=5\left(x-1\right)\\ y-2=5x-5\\ y=5x-5+2\\ y=5x-3\end{array}$

Substitute $y=0$ in equation $y=5x-3$ to obtain the x intercept as follows:

$\begin{array}{c}0=5x-3\\ 3=5x\\ x=\frac{5}{3}\end{array}$

Therefore, the $x$ -intercept of the line that passes through the points $\left(1,2\right)$ and $\left(2,7\right)$ is $\frac{3}{5}$ .

The answer matches Option C.

Thus, the correct option is C.