Chapter 1.3, Problem 56E

To determine

To find an equation of the line passing through the pair of given points and sketch the line.

Answer to Problem 56E

An ordered pair $\left(a,b\right)$ is a solution of an equation in x and y when substitute $x=a\text{and}y=b$ .

Explanation of Solution

Given information:

The points are $\left(4,3\right)\text{and}\left(-4,-4\right)$ .

Equation of line passes through two points.

Let two points are $\left(x_1,y_1\right)\text{and}\left(x_2,y_2\right)$ , then the equation of the line is,

  $y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$ .

Calculation:

Let two points are $\left(x_1,y_1\right)=\left(4,3\right)\text{and}\left(x_2,y_2\right)=\left(-4,-4\right)$

Then substitute these points in the above-mentioned equation of line.

  $\begin{array}{c}y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)\\ y-3=\frac{-4-3}{-4-4}\left(x-4\right)\\ y-3=\frac{-7}{-8}\left(x-4\right)\\ y-3=\frac{7}{8}x-4\cdot \frac{7}{8}\\ y-3=\frac{7}{8}x-\frac{7}{2}\end{array}$

Add 3 from both sides.

  $\begin{array}{c}y-3+3=\frac{7}{8}x-\frac{7}{2}+3\\ y=\frac{7}{8}x+\frac{-7+6}{2}\\ y=\frac{7}{8}x-\frac{1}{2}\end{array}$

So, the equation of the line is $y=\frac{7}{8}x-\frac{1}{2}$ .

The value of y at the certain points,

x	-4	-2	0	0.571	2	3
y	-4	-2.25	0.5	0	1.25	2.125

The graph of the line $y=\frac{7}{8}x-\frac{1}{2}$ is shown below.