Question

Without graphing, determine the number of solutions and then classify the system of equations.

$$\begin{gathered} y=\frac{1}{3}x-5 \\ x-3y=6 \end{gathered}$$

Short answer

The system of equations has no solution and is inconsistent and independent.

Step-by-step solution

Step 1. Given information 

The system of equations is

$$\begin{gathered} y=\frac{1}{3}x-5 \\ x-3y=6 \end{gathered}$$

Step 2. Find the slope and intercept of the first equation.

The first equation is

$y=\frac{1}{3}x-5$

Compare with the slope-intercept formula of line

$y=mx+b$

So,

$$\begin{gathered} m=\frac{1}{3} \\ b=-5 \end{gathered}$$

Step 3. Find the slope and intercept of the second equation.

The second equation is

$x-3y=6$

Solve for $y$

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

Compare with the slope-intercept formula of line

$y=mx+b$

So,

$$\begin{gathered} m=\frac{1}{3} \\ b=-2 \end{gathered}$$

Step 4. Compare the slopes and intercepts of the two lines.

First equation:

Slope is $\frac{1}{3}$

y-intercept is $-5$

Second equation:

Slope is $\frac{1}{3}$

y-intercept is $2$

Since the slopes are the same and y-intercepts are different.

So, the lines are parallel.

A system of equations whose graphs are parallel lines has no solution and is inconsistent and independent.