Chapter 1.3, Problem 34E

To determine

To find: the value of a, if $C\left(a,1\right)$ is on the line which passes through $A\left(3,7\right)$ and $B\left(-4,9\right)$ .

Answer to Problem 34E

24

Explanation of Solution

Given information:

The three points on a line are as follows:

  $\begin{array}{l}A\left(3,7\right)\\ B\left(-4,9\right)\\ C\left(a,1\right)\end{array}$

Calculation:

Use the slope intercepts formula to find the slope and intercept of line.

  $y=mx+b$

To find the slope, use slope formula given below.

  $m=\frac{y_2-y_1}{x_2-x_1}$

Put the coordinates from points A and B in the slope formula and evaluate the slope as follows:

  $\begin{array}{c}m=\frac{9-7}{-4-3}\\ =-\frac{2}{7}\end{array}$

Thus, the slope is $-\frac{2}{7}$ .

Now, put the value of slope, and the coordinates of point A, in the slope intercept formula to calculate b as shown below.

  $\begin{array}{c}y=mx+b\\ 7=-\frac{2}{7}\left(3\right)+b\\ 7=-\frac{6}{7}+b\\ b=7+\left(\frac{6}{7}\right)\\ b=\frac{49+6}{7}\\ =\frac{55}{7}\end{array}$

Thus, the intercept is $\frac{55}{7}$ .

Hence, the slope-intercept equation for a line that passes through A, B and C can be written as follows:

  $y=\left(-\frac{2}{7}\right)x+\frac{55}{7}$ … (1)

Now substitute the coordinates of point $C\left(a,1\right)$ in equation (1) and evaluate a as shown below.

  $\begin{array}{c}1=\left(-\frac{2}{7}\right)\left(a\right)+\frac{55}{7}\\ \frac{2}{7}a=\frac{55}{7}-1\\ \frac{2}{7}a=\frac{55-7}{7}\\ \frac{2}{7}a=\frac{48}{7}\\ a=\frac{48}{7}\left(\frac{7}{2}\right)\\ =24\end{array}$

Thus, the value of a is 24.