Q. 4.11 - Intermediate Algebra

Question

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

$$\begin{gathered} \text{ (a) }\left\{\begin{array}{l}y=-2x-4\\ 4x+2y=9\end{array}\right. \\ \text{ (b) }\left\{\begin{array}{l}3x+2y=2\\ 2x+y=1\end{array}\right. \end{gathered}$$

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 $(a) \left\{\begin{array}{l}y=-2x-4\\ 4x+2y=9\end{array}\right.$

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

Since the first equation is already in slope-intercept form. $y=-2x-4$

Write the second equation in slope-intercept form $4x+2y=9$

$\begin{array}{rcl}2y& =& -4x+9\\ y& =& -\frac{4}{2}x+\frac{9}{2}\\ & =& -2x+4.5\end{array}$

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

First equation in slope-intercept form$y=-2x-4$

Slope $m=-2$

$y-$intercept $b=-4$

The slope-intercept form of the second equation is $y=-2x+4.5$.

Slope $m=-2$

$y-$intercept $b=4.5$

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 a system of equations.

$(b) \left\{\begin{array}{l}3x+2y=2\\ 2x+y=1\end{array}\right.$

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

The first equation is in slope-intercept form.

$\begin{array}{rcl}3x+2y& =& 2\\ 2y& =& -3x+2\\ y& =& -\frac{3}{2}x+\frac{2}{2}\\ & =& -1.5x+1\end{array}$

The second equation is in slope-intercept form.

$\begin{array}{rcl}2x+y& =& 1\\ y& =& -2x+1\end{array}$

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

The first equation in slope-intercept form: $y=-1.5x+1$

Slope $m=-1.5$

$y-$intercept $b=1$

The second equation in slope-intercept form: $y=-2x+1$

Slope $m=-2$ and $y-$intercept $b=1$

• Since the slopes are different, the lines intersect.
• The number of solutions of the equations is 1.