Chapter 4.3, Problem 34PPS

To determine

Whether the pair of equations is perpendicular, parallel or neither for equations $y=-2x$ and $2x+y=3$ .

Answer to Problem 34PPS

The equation is parallel.

Explanation of Solution

Given:

The Pair of equations are $y=-2x$ and $2x+y=3$ .

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 $y=-2x$ ,

$y=-2x\mathrm{......}\left(1\right)$

On comparison with standard equation, the slope for equation (1) is $m_1=-2$ .

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

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

On comparison with standard equation, the slope for equation (2) is $m_2=-2$ .

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

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

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

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

  $\frac{-2}{2}=\frac{-1}{1}\ne \frac{0}{3}$

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