Chapter 4.3, Problem 33PPS

To determine

Whether the pair of equations is perpendicular, parallel or neither.

Answer to Problem 33PPS

The equation is neither parallel nor perpendicular.

Explanation of Solution

Given:

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

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 $y=4x+3$ ,

$y=4x+3\mathrm{......}\left(1\right)$

On comparison with standard equation, the slope for equation (1) is $m_1=4$ .

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

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

On comparison with standard equation, the slope for equation (2) is $m_1=-4$ .

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

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

$\begin{array}{c}{m}_1{m}_2=4\left(-4\right)\\ =-16\end{array}$

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) $\left(-4x+y-3=0\right)$ and (2) $\left(4x+y-3=0\right)$ ,

  $\frac{-4}{4}\ne \frac{1}{1}=\frac{-3}{-3}$

Thus, the equation is neither parallel nor perpendicular. The graph for both the equations is shown below: