Chapter 4.3, Problem 36PPS

To determine

Whether the pair of equations is perpendicular, parallel or neither for equations $-3x+4y=8$ and $-4x+3y=-6$ .

Answer to Problem 36PPS

The pair of equations is neither perpendicular nor parallel.

Explanation of Solution

Given:

The Pair of equations are $-3x+4y=8$ and $-4x+3y=-6$ .

Calculation:

Standard equation of slope intercept form is expressed as follows:

  $y=mx+c$

Here, m is slope and c stands for y -intercept.

For equation $-3x+4y=8$ ,

  $\begin{array}{l}-3x+4y=8\\ y=\frac{3}{4}x+2\mathrm{......}\left(1\right)\end{array}$

On comparison with standard equation, the slope for equation (1) is $m_1=\frac{3}{4}$ .

Separate the y term of the second equation for the slope-intercept form as follows:

  $\begin{array}{c}-4x+3y=-6\\ y=\frac{4}{3}x-2\mathrm{......}\left(2\right)\end{array}$

On comparison with standard equation, the slope for equation (2) is $m_2=\frac{4}{3}$ .

Any pair of lines is perpendicular if their multiplication of slopes is -1.

Multiply the slopes of equations (1) and (2) as follows:

  $\begin{array}{c}{m}_1{m}_2=\left(\frac{3}{4}\right)\left(\frac{4}{3}\right)\\ =1\end{array}$

Thus, the lines are not perpendicular.

For parallel condition,

Coefficient ratio must be $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$ .

For equation (1) $\left(-3x+4y=8\right)$ and (2) $\left(-4x+3y=-6\right)$ ,

  $\frac{-3}{-4}\ne \frac{4}{3}\ne \frac{-8}{6}$

Thus, the equation is not parallel. The graph for both the equations is shown below: