Chapter 1.1, Problem 30E

To determine

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

Answer to Problem 30E

Three possible points on the given line is $(1,20),(-1,14)\text{ and }(2,23)$ .

Explanation of Solution

Given Information:

The point in a line with slope $m=3$ is $\left(-4,5\right)$

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 xand 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 $\left(-4,5\right)$ and slope $m=3$ in (1), the equation of line can be found as,

  $\begin{array}{l}y-\left(5\right)=3(x-\left(-4\right))\\ y-5=3x+12\\ y=3x+17\end{array}$

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,

  $\begin{array}{c}y=3\left(1\right)+17\\ =20\end{array}$

Thus, the point is $(1,20)$

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

  $\begin{array}{c}y=3\left(-1\right)+17\\ =14\end{array}$

Thus, the coordinates of the point is $(-1,14)$

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

  $\begin{array}{c}y=3\left(2\right)+17\\ =23\end{array}$

Thus, the coordinates of the point is $(2,23)$