Question

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

$2x-3y=10;$width="102">3y−2x−7=0

Short answer

Given lines areparallelsince they have the same slope.

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 2x-3y=10.

We have$2x-3y=10$

Subtract 2x to the both sides:

$\begin{array}{l}2x-3y-2x=10-2x\\ -3y=-2x+10\end{array}$

Divide both sides by $-3$:

$\begin{array}{l}\frac{-3y}{-3}=\frac{-2x+10}{-3}\\ y=\frac{2}{3}x-\frac{10}{3}\end{array}$

Compare with slope-intercept form, we get

$m_1=\frac{2}{3}$

Step 3. Find the slope of 3y-2x-7=0.

We have $3y-2x-7=0$

Add 7 to the both sides:

$\begin{array}{l}3y-2x-7+7=0+7\\ 3y-2x=7\end{array}$

Add 2x to the both sides of the equation:

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

Divide by 3 on the both sides:

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

Compare with slope-intercept form, we get

$m_2=\frac{2}{3}$

So, both the given lines are parallel since they have the same slope.