Chapter 13.9, Problem 3ST2

To determine

To calculate: The value of slope and $y$ -intercept to create an equation using given two points on the line.

Answer to Problem 3ST2

The equation of the line through the points $(1,2)$ and $(5,0)$ is $x+2y=5$ .

Explanation of Solution

Given information: The points on the line are $(1,2)$ and $(5,0)$ .

Calculation:

The slope formula, when two points $(x_{1,}{y}_1)$ and $(x_{2,}{y}_2)$ are plotted on a line:

  $m=\frac{y_2-y_1}{x_2-x_1}$

Let, $(x_1,y_1)$$=(1,2)$ and $(x_{2,}{y}_2)=(5,0)$

  $\therefore m=\frac{0-2}{5-1}=\frac{-2}{4}=-\frac{1}{2}$

According to the slope-intercept form of a linear equation $y=mx+c$ ,

  $y=-\frac{1}{2}x+c$

Now, choose any one of the two points as the values of $x$ and $y$ because in either case the answer will be same.

Let, $x$$=$$x_1$$=$ 1 and $y$$=$$y_1$$=$$2$

  $\begin{array}{l}\therefore 2=-\frac{1}{2}\cdot 1+c\\ 2=-\frac{1}{2}+c\\ c=2+\frac{1}{2}=\frac{4+1}{2}=\frac{5}{2}\end{array}$

Now, put the value of slope, $m$$=$$-\frac{1}{2}$ and the value of $y$ -intercept, $c$$=$$\frac{5}{2}$ in the equation $y=mx+c$ ,

  $\begin{array}{l}y=-\frac{1}{2}x+\frac{5}{2}=\frac{-x+5}{2}\\ y=\frac{5-x}{2}\\ 2y=5-x\\ x+2y=5\end{array}$

Hence, the equation of the line through $(1,2)$ and $(5,0)$ is $x+2y=5$ .