Chapter 6.2, Problem 14P

Reasoning Can you tell that there is no solution for a system by just looking at the equations? Explain and give an example.

To determine

To find:the condition satisfied by the system of equations so that the system has no solution.

Answer to Problem 14P

Yes, we can tell that there is no solution for a system by just looking at the equations.

For example:

  $\begin{array}{l}2x+3y-12=0\\ 4x+6y-36=0\end{array}$

Explanation of Solution

First, we will write the condition for no solution.

Consider the following equations:

  $\begin{array}{l}{a}_{1}x+b_{1}y+c_1=0\\ a_{2}x+b_{2}y+c_2=0\end{array}$

The system has no solution if $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$

Now we will give an example.

For example, $2x+3y-12=0,\text{}\text{}4x+6y-36=0$

First, we will try to find the solution of the above equations

  $\begin{array}{l}2x+3y-12=0(i)\\ 4x+6y-36=0(ii)\end{array}$

Multiply $(i)$ by 2

  $4x+6y-24=0(iii)$

Subtract $(ii)$ from $(iii)$

  $\begin{array}{l}\left(4x+6y-24\right)-\left(4x+6y-36\right)=0\\ \Rightarrow -24+36=0\end{array}$

  $\Rightarrow 12=0$ which is not true.

So, the system of equations has no solution.

Now we will check the condition $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$

For equations $2x+3y-12=0,\text{}\text{}4x+6y-36=0$ ,

  $\begin{array}{l}{a}_1=2,b_1=3,c_1=-12\\ a_2=4,b_2=6,c_2=-36\end{array}$

Now find $\frac{a_1}{a_2},\frac{b_1}{b_2},\frac{c_1}{c_2}$

  $\frac{a_1}{a_2}=\frac{2}{4}=\frac{1}{2},\frac{b_1}{b_2}=\frac{3}{6}=\frac{1}{2},\frac{c_1}{c_2}=\frac{-12}{-36}=\frac{1}{3}$

Therefore,

  $\frac{a_1}{a_2}\left(\frac{1}{2}\right)=\frac{b_1}{b_2}\left(\frac{1}{2}\right)\ne \frac{c_1}{c_2}\left(\frac{1}{3}\right)$

So, the given conditions for no solution are also satisfied.