Chapter 3, Problem 2CT

Solution

To identify: The system without graphing. Then find the solution to each system using a graph.

  $\left\{\begin{array}{c}3x+2y=9\\ 3x+2y=4\end{array}\right.$

The system has no solutions.

Given information:

The given system is:

  $\left\{\begin{array}{c}3x+2y=9\\ 3x+2y=4\end{array}\right.$

Explanation:

For two equations $a_{1}x+b_{1}y+c_1=0$ and $a_{2}x+b_{2}y+c_2=0$ .

The system of equation will be consistent but independent with one solution if: $\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$

The system of equation will be consistent but dependent with infinitely many solutions if: $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

The system of equation will be inconsistent with no solution if: $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$

Now consider the given system:

  $\left\{\begin{array}{c}3x+2y=9\\ 3x+2y=4\end{array}\right.$

The equations can be written as:

  $\begin{array}{c}3x+2y-9=0\\ 3x+2y-4=0\end{array}$

Now check if the system of equations is independent, dependent, or inconsistent:

  $\frac{3}{3}=\frac{2}{2}\ne \frac{-9}{-4}$

It can also be written as:

  $\frac{1}{1}=\frac{1}{1}\ne \frac{-9}{-4}$

The system of equation will be inconsistent with no solution

Therefore, the lines will not intersect nowhere.

Now consider the equation $3x+2y=9$ :

For $x=0$ ,

  $\begin{array}{c}3\left(0\right)+2y=9\\ 2y=9\\ y=\frac{9}{2}\end{array}$

For $y=0$ ,

  $\begin{array}{c}3x+2\left(0\right)=9\\ 3x=9\\ x=3\end{array}$

Now consider the equation $3x+2y=4$ :

For $x=0$ ,

  $\begin{array}{c}3\left(0\right)+2y=4\\ 2y=4\\ y=2\end{array}$

For $y=0$ ,

  $\begin{array}{c}3x+2\left(0\right)=4\\ 3x=4\\ x=\frac{4}{3}\end{array}$

Now graph each equation:

  

From the graph the system has no solutions.