Chapter 2.1, Problem 20E

To determine

To find: the system of given equations by graphing.

Answer to Problem 20E

Consistent and independent

Explanation of Solution

Given information:

  $\begin{array}{l}3x-8y=10\mathrm{...}\left(1\right)\\ 16x-32=75\mathrm{...}\left(2\right)\end{array}$

Calculation:

Convert equation (1) into slope intercept form to evaluate the value of slope and y-intercept as follows:

  $\begin{array}{c}3x-8y=10\\ -8y=-3x+10\\ y=\frac{-3}{-8}x+\frac{10}{-8}\\ y=\frac{3}{8}x-\frac{5}{4}\end{array}$

Thus, the value of slope is $\frac{3}{8}$ and y-intercept is $-\frac{5}{4}$ .

Convert equation (2) into slope intercept form to evaluate the value of slope and y-intercept as follows:

  $\begin{array}{c}16x-32y=75\\ -32y=-16x+75\\ y=\frac{-16}{-32}x+\frac{75}{-32}\\ y=\frac{1}{2}x-\frac{75}{32}\end{array}$

Thus, the value of slope is $\frac{1}{2}$ and y-intercept is $-\frac{75}{32}$ .

Since, both the slopes have different values and the lines are not parallel to each other. So, they can have only one solution.

So, without graphing, the equations are consistent and independent.