Chapter 2.6, Problem 61E

To determine

Tofind:The slope − intercept form of equation through given points and to graph the line.

Answer to Problem 61E

The general form of equation of line passing through points $\left(2,7\right)$ and $\left(2,10\right)$ is $ax+0\cdot y=c$ or

  $x=2$ .The graph of line $x=2$ is shown in Figure-1.

Explanation of Solution

Given information:

The required line passes through points $\left(2,7\right)$ and $\left(2,10\right)$ . If slope-intercept form is not possible then explain why and use general form.

Conceptused:

The slope intercept form of the line is $y=mx+c$ , where m is the slope of the line and c the intercept made by the line on y -axis.

Calculations:

Suppose the slope-intercept form of the line passing through points $\left(2,7\right)$ and $\left(2,10\right)$ is

  $y=mx+c$    … (1)

The point $\left(2,7\right)$ is on the line (1) so it must satisfy the equation $y=mx+c$

  $7=m\cdot 2+c$

  $\Rightarrow 2m+c=7$    … (2)

The point $\left(2,10\right)$ is on the line (1) so it must satisfy the equation $y=mx+c$

  $10=m\cdot 2+c$ .

  $\Rightarrow 2m+c=10$    … (3)

We observe that equations (2) and (3) cannot be solved.

As the solution of equation (2) and (3 is possible so we use general $ax+by=c$ .

The point $\left(2,7\right)$ is on the line (1) so it must satisfy the equation $ax+by=c$

  $=a\cdot 2+b\cdot 7=c$

  $\Rightarrow 2a+7b=c$    … (4)

The point $\left(2,10\right)$ is on the line (1) so it must satisfy the equation $ax+by=c$

  $=a\cdot 2+b\cdot 10=c$

  $\Rightarrow 2a+10b=c$    … (5)

Now, subtracting equation (4) from equation (5), we have $3b=0\Rightarrow b=0$ , and $2a=c$

Hence, the general form of equation of line passing through points $\left(2,7\right)$ and $\left(2,10\right)$ is $ax+0\cdot y=c$

  $x=\frac{c}{a}$

  $\Rightarrow x=\frac{2a}{a}=2$

The graph of line $x=2$ is shown in Figure-1 here.

  

Conclusion:

The general form of equation of line passing through points $\left(2,7\right)$ and $\left(2,10\right)$ is $ax+0\cdot y=c$ or

  $x=2$ .The graph of line $x=2$ is shown in Figure-1.