Question

The equations of two lines are given. Determine whether the lines are parallel, perpendicular, or neither. (Lesson 1.10)

$7x-3y=2;$$9y+21x=1$

Short answer

Given lines areneither parallel nor perpendicular.

Step-by-step solution

Step 1. Definition of the slope-intercept form.

Point-intercept is the equation of a straight line in the form$y=mx+b$, where m is the slope and b is y-intercept.

Step 2. Find the slope of 7x-3y=2.

We have$7x-3y=2$

Subtract 7xto the both sides of the equation:

$\begin{array}{l}7x-3y-7x=2-7x\\ -3y=-7x+2\end{array}$

Divide both sides by $-3$:

$\begin{array}{l}\frac{-3y}{-3}=\frac{-7x+2}{-3}\\ y=\frac{7}{3}x-\frac{2}{3}\end{array}$

Compare with slope-intercept form, we get

$m_1=\frac{7}{3}$

Step 3. Find the slope of 9y+21x=1.

We have$9y+21x=1$

Subtract 21x to the both sides of the equation:

$\begin{array}{l}9y+21x-21x=1-21x\\ 9y=-21x+1\end{array}$

Divide both sides by 9:

$\begin{array}{l}\frac{9y}{9}=\frac{-21x+1}{9}\\ y=-\frac{7}{3}x+\frac{1}{9}\end{array}$

Compare with slope-intercept form, we get

$m_2=-\frac{7}{3}$

So, both the given lines are neither parallel nor perpendicular since their slopes are neither equal nor negative reciprocals of each other.