Chapter 1.1, Problem 31E

To determine

To find: Three additional points on a line with a given point and slope

Answer to Problem 31E

Three possible points on the given line are $(1,-5)$ , $(-1,-6)$ , $\left(2,-\frac{9}{2}\right)$ .

Explanation of Solution

Given Information:

The point in a line with slope $m=\frac{1}{2}$ is $(7,-2)$

Formula used:

When the slope of the line and a point on the line is given, the equation of line can be obtained using the point-slope form as,

  $y-y_1=m(x-x_1)$ ......(1)

The equation of a line is of the form $y=mx+c$ where m denotes the slope, $(x,y)$ denotes the x and y coordinates in the line and c is a constant. So, by choosing a value for $x=0,\pm 1,\pm \mathrm{2..}$ the corresponding value of y can be obtained which gives the coordinates of different points on the line.

Calculation:

Substituting the given point $(7,-2)$ and slope $m=\frac{1}{2}$ in (1), the equation of line can be found as,

  $\begin{array}{l}y-\left(-2\right)=\frac{1}{2}(x-7)\\ y+2=\frac{1}{2}x-\frac{7}{2}\\ y=\frac{1}{2}x-\frac{7}{2}-2\\ y=\frac{1}{2}x-\frac{11}{2}\end{array}$

Thus, the equation of the line is obtained. Now, three different points on the line can be found as follows.

Consider $x=1$ . Substituting it in the equation of the line,

  $\begin{array}{c}y=\frac{1}{2}\left(1\right)-\frac{11}{2}\\ =-5\end{array}$

Thus, the point is $(1,-5)$

Now, consider $x=-1$ . Substituting it in the equation of the line gives,

  $\begin{array}{c}y=\frac{1}{2}\left(-1\right)-\frac{11}{2}\\ =-6\end{array}$

Thus, the coordinates of the point is $(-1,-6)$

Now, considering $x=2$ . Substituting it in the equation of the line gives,

  $\begin{array}{c}y=\frac{1}{2}\left(2\right)-\frac{11}{2}\\ =-\frac{9}{2}\end{array}$

Thus, the coordinates of the point is $\left(2,-\frac{9}{2}\right)$ .