Question

The pair of equations x = 0 and y = -7 has
A. One solution.
B. Two solutions.
C. Infinitely many solutions.
D. No solution.

Answer 1

Hint: In this question we will use the method of finding the solutions of pairs of linear equations in two variables. Here we have given two equations in the question. We will solve these equations and find out the solvability of these equations that either they have one solution or two solutions or many solutions.

Complete step-by-step solution -
 We know that if,
        \[{a_1}x + {b_1}y + {c_1} = 0\] and …………(i)
        \[{a_2}x + {b_2}y + {c_2} = 0\] ………….(ii)
 is a pair of linear equations in two variables x and y such that
(i) $\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$, then the pair of linear equations is consistent with a unique solution.
(ii) $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}$, then the pair of linear equations is inconsistent.
(iii) $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}$, then the pair of linear equations is consistent with infinitely many solutions.
Here we have, x = 0 and y = -7. We can also write these equations as:
$1x + 0y + 0 = 0 $ ………..(iii)
$0x + 1y + 7 = 0 $ …………(iv)
Now comparing equation (iii) with equation(i) and equation(iv) with equation(ii), we get
${a_1} = 1,{b_1} = 0$ and ${c_1} = 0$ , and ${a_2} = 0,{b_2} = 1$ and ${c_2} = 7$.
Here we have,
$\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{1}{0} = \infty ,{\text{ }}\dfrac{{{b_1}}}{{{b_2}}} = \dfrac{0}{1} = 0{\text{ and }}\dfrac{{{c_1}}}{{{c_2}}} = \dfrac{0}{7} = 0$.
Now according to the method stated above we can see that,
$ \Rightarrow $ $\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$, hence we can say that the pair of linear equations is consistent with a unique solution (one solution).
Therefore, the correct answer is option (A).

Note : In this type of question ,we have to find out the equations given in the question and then we have to compare those linear equations with the general form of a pair of linear equations. By comparing we can easily get the terms like ${a_1},{b_1},{c_1},{a_2},{b_2}$and ${c_2}$. After that we will use the conditions of finding the solutions of a pair of linear equations. Through this we will get the required answer.