Question

Check whether the following pair of linear equations are consistent or inconsistent.
3x + 2y = 5, 2x - 3y = 7.

Answer 1

Hint: For checking whether the pair of linear equations are consistent or inconsistent, we try to obtain values of x and y. If both x and y have a unique value then the system is consistent. The system becomes inconsistent when there exist no values of x and y that satisfy both the equations.

Complete step-by-step answer:
According to the given system of equations, we assign equations corresponding to the expression.
Let the first expression be: $3x+2y=5\ldots (1)$
The second expression will be: $2x-3y=7\ldots (2)$
Now, we try to eliminate one of the variables x or y by using both the equations.
To do so, we multiply the equation (1) with 3 and multiply the equation (2) with 2.
$\begin{aligned} & \left( 3x+2y=5 \right)\times 3 \\\ & 9x+6y=15\ldots (3) \\\ & \left( 2x-3y=7 \right)\times 2 \\\ & 4x-6y=14\ldots (4) \\\ \end{aligned}$
Since both the equations have the same value of y, it can be eliminated. Now, adding equation (3) and (4), we get
$\begin{aligned} & 9x+6y-15+\left( 4x-6y-14 \right)=0 \\\ & 9x+4x+6y+6y-15-14=0 \\\ & 13x-29=0 \\\ & x=\dfrac{29}{13} \\\ \end{aligned}$
So, the obtained value of x is $\dfrac{29}{13}$.
Putting the value of x in equation 1, we get
$\begin{aligned} & 3\times \dfrac{29}{13}+2y=5 \\\ & 2y=5-\dfrac{87}{13} \\\ & 2y=\dfrac{65-87}{13} \\\ & 2y=-\dfrac{22}{13} \\\ & y=-\dfrac{11}{13} \\\ \end{aligned}$
Hence, the value of y is $-\dfrac{11}{13}$.
Since there exists a unique value of x and y, therefore the system is consistent.

Note: This problem can alternatively be solved by using the coefficient analysis method for determination of consistent system. In this method the coefficients of x and y i.e. a and b, are compare and if the condition $\dfrac{{{a}_{1}}}{{{a}_{2}}}\ne \dfrac{{{b}_{1}}}{{{b}_{2}}}$ is satisfied, then the system is consistent.