Q. 4.12 - Intermediate Algebra

Question

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

$$\begin{gathered} \text{(a) }\left\{\begin{array}{l}y=\frac{1}{3}x-5\\ x-3y=6\end{array}\right. \\ (b) \left\{\begin{array}{l}x+4y=12\\ -x+y=3\end{array}\right. \end{gathered}$$

Step-by-Step Solution

Answer

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

Part (b) A system of equations whose graphs are intersected has 1 solution and is consistent and independent.

Part (a) Step 1. Given Information.

We have been given system of equations:

 $\text{(a) }\left\{\begin{array}{l}y=\frac{1}{3}x-5\\ x-3y=6\end{array}\right.$

Part (a) Step 2. Writing both equations in slope-intercept form.

The first equation is already in slope-intercept form $y=\frac{1}{3}x-5$

Write the second equation in slope-intercept form,

$x-3y=6$

$3y=x-6$

$y=\frac{1}{3}x-\frac{6}{3}$

$y=\frac{1}{3}x-2$

Part (a) Step 3. Find the slope and intercept of each line.

The first equation slope-intercept form:$y=\frac{1}{3}x-5$

Slope $m=\frac{1}{3}$ and $y-$intercept $b=-5$

The second equation slope-intercept form:$y=\frac{1}{3}x-2$

Slope $m=\frac{1}{3}$ and $y-$intercept $b=-2$

Since the slopes are the same and $y-$intercepts are different, the lines are parallel.

The number of solutions of the equations is 0

Part (b) Step 1. Given Information.

We have been given system of equations:

$(b) \left\{\begin{array}{l}x+4y=12\\ -x+y=3\end{array}\right.$

Part (b) Step 2. Writing both equations in slope-intercept form.

First equation in slope-intercept form.

$x+4y=12$

$4y=-x+12$

$y=-\frac{1}{4}x+3$

The second equation slope-intercept form: 

$-x+y=3$

        $y=x+3$

Part (b) Step 3. Find the slope and intercept of each line.

The first equation slope-intercept form:$y=-\frac{1}{4}x+3$

Slope $m=-\frac{1}{4}$ and $y-$intercept $b=3$

The second equation slope-intercept form:$y=x+3$

Slope $m=1$ and $y-$intercept $b=3$

Since the slopes are different, the lines intersect. 

The number of solutions of the equations is 1.