Question

The number of solutions of the equations $2x - 3y = 5,x + 2y = 7$ is …..
A) 1
B) 2
C ) 4
D) 0

Answer 1

Hint: To find the number of solutions of the linear equations in two variables we discuss nature of the equations. If the nature of the equations is consistent then equations will have one solution, if inconsistent then equations will have zero solution, if coincident then equations will have infinite solutions.
Formulas used: Consistent Solution $\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$ , Inconsistent $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}$ and Coincident $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}$

Complete Step by Step Solution

We first compare given linear equations with standard linear equations to get coefficients ${a_1},{b_1},{c_1},{a_2},{b_2}$ and ${c_2}$.
Standard linear equations are ${a_1}x + {b_1}y = {c_1}$ and${a_2}x + {b_2}y = {c_2}$. Given equations are $2x - 3y = 5$ and $x + 2y = 7$

On comparing we have ${a_1} = 2,\,{b_1} = - 3,\,{c_1} = 5$ and ${a_2} = 1,\,{b_2} = 2,\,{c_2} = 7$

Conditions to find number of solutions is:
For Consistent (or one solution) $\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$
For Inconsistent (zero solution) $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}$
For Coincident (infinite solution) $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}$

Now, to discuss the number of solutions given linear equations, have we put the values in the following conditions to see which condition values will satisfy?
$\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{2}{1}$ , and $\dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{ - 3}}{2}$
Clearly from above we see that $\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$
Therefore, there is no need to check further conditions as the given linear equation satisfies the first condition.

Therefore, we can say that given linear equations have inconsistent solutions or we can say only one solution.
Hence, the correct option is (A).


Note: We can also use a method of elimination to find the number of solutions of the given linear equations. If on solving given equations by elimination method we get 1 solution then option (A) will be the right option, but if we are unable to solve equations then option (D) will be the right option.