Question

Determine whether the graphs of the pairs of equations are parallel, perpendicular, or neither.

$\begin{array}{l}y=-6x+8\\ 3x+\frac{1}{2}y=-3\end{array}$

F parallel

G perpendicular

H neither

J not enough information

Short answer

The equation in slope-intercept form is$\mathit{y}\mathbf{=}\mathbf{4}\mathit{x}\mathbf{+}\mathbf{15}$.

Step-by-step solution

Step 1. State the concept of slope-intercept of an equation and slope of parallel and perpendicular lines.

The slope-intercept form of a linear equation is $y=mx+b$, where m is the slope and $b$is the $y$-intercept.

Parallel lines: Slopes of the parallel lines are the same.

Perpendicular lines: The slopes of the nonvertical perpendicular lines are opposite reciprocals.

Step 2. Calculate the slopes of the two equations. 

Calculate the slope of the first equation. The first equation is in the form of slope and intercept form. Compared to the standard form of the equation $y=mx+b$, the slope of the first equation is:

$m=-6$

Calculate the slope of the second equation. Convert the second equation in the slope-intercept form.

$$\begin{gathered} 3x+\frac{1}{2}y=-3 \\ \frac{1}{2}y=-3x-3 \\ y=-6x-6 \end{gathered}$$

Compared to the standard form of the equation $y=mx+b$, the slope of the second equation is:

$m=-6$

Step 3. Compare the slopes of the equations.

The slope of both the equation is$m=-6$ which issame and equal. Thus, both the equations are parallel.

Therefore, option F is correct. The pair of equations are parallel.