Question

On comparing the ratios$\frac{a_1}{a_2},\frac{b_1}{b_2},\frac{c_1}{c_2}$, find out whether the following pairs of linear equations are consistent or inconsistent:

$$\begin{gathered} 3x+2y=5 \\ 2x-3y=7 \end{gathered}$$

Answer 1

Step 1: Analyzing the condition by which lines are intersecting, parallel or coincident :

If we are given a set of two linear equations,

$a_{1}x+b_{1}y+c_1=0$

$a_{2}x+b_{2}y+c_2=0$ , then

• lines are consistent, if $\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$ and $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$
• lines are inconsistent if $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$

Step 2: Verify the given set of the equations:

The equations are -

$$\begin{gathered} 3x+2y=5 \\ 2x-3y=7 \end{gathered}$$

By comparing the given equations with

$a_{1}x+b_{1}y+c_1=0$

$a_{2}x+b_{2}y+c_2=0$,

$a_1=3,a_2=2,b_1=2,b_2=-3,c_1=5,c_2=7$

$$\begin{gathered} \frac{a_1}{a_2}=\frac{3}{2}, \\ \frac{b_1}{b_2}=\frac{2}{-3} \\ \frac{c_1}{c_2}=\frac{5}{7} \end{gathered}$$

Since, $\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$, therefore the given equations intersect each other at one point and they have only one possible solution.

Hence, the given set of linear equations are consistent.