Chapter 4.3, Problem 38PPS

To determine

Whether the pair of equations is perpendicular, parallel or neither for equations $2x+7y=-35$ and $4x+14y=-42$ .

Answer to Problem 38PPS

The pair of equations is parallel.

Explanation of Solution

Given:

The Pair of equations are $2x+7y=-35$ and $4x+14y=-42$ .

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 $2x+7y=-35$ ,

  $\begin{array}{l}2x+7y=-35\\ y=-\frac{2}{7}x-5\mathrm{......}\left(1\right)\end{array}$

On comparison with standard equation, the slope for equation (1) is $m_1=-\frac{2}{7}$ .

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

  $\begin{array}{c}4x+14y=-42\\ y=-\frac{2}{7}x-3\mathrm{......}\left(2\right)\end{array}$

On comparison with standard equation, the slope for equation (2) is $m_2=-\frac{2}{7}$ .

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

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

  $\begin{array}{c}{m}_1{m}_2=\left(-\frac{2}{7}\right)\left(-\frac{2}{7}\right)\\ =\frac{4}{49}\end{array}$

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) $\left(2x+7y=-35\right)$ and (2) $\left(4x+14y=-42\right)$ ,

  $\frac{2}{4}=\frac{7}{14}\ne \frac{-35}{-42}$

Thus, the equation is parallel. The graph for both the equations is shown below: