Question

Find three solutions of the equation: $ 2x + y = 7 $
(A) $ \left( {0,7} \right),\left( {\dfrac{7}{2},0} \right){\rm{ and }}\left( {3,1} \right) $
(B) $ \left( {0,6} \right),\left( {\dfrac{7}{2},0} \right){\rm{ and }}\left( {3,1} \right) $
(C) $ \left( {0,7} \right),\left( {\dfrac{5}{2},0} \right){\rm{ and }}\left( {3,1} \right) $
(D) $ \left( {0,7} \right),\left( {\dfrac{7}{2},0} \right){\rm{ and }}\left( {2,1} \right) $

Answer 1

Hint: In this question one linear equation having two variables is given. We have to solve this equation and find three solutions. In order to solve this question, we assume any arbitrary value of either of the variables and put it in the equation, so we get one solution. Then repeat this step and put another arbitrary value into the equation and obtain the other solutions as well.

Complete step-by-step answer:
Given:
The linear equation having variables $ x $ and $ y $ is given as –
 $ 2x + y = 7 $
For the initial solution of the equation, put $ x = 0 $ in the linear equation and obtain the value of variable $ y $ .
So, substituting $ x = 0 $ in the equation, we get,
 $
2 \times 0 + y = 7\\
y = 7
 $
So, the first solution of the linear equation is
 $ x = 0 $
 $ y = 7 $
In two-dimensional coordinates form $ \left( {x,y} \right) $ , the first solution is $ \left( {0,7} \right) $ .
Now, similarly,
Substituting $ y = 0 $ in the linear equation we have,
 $
\Rightarrow 2x + 0 = 7\\
\Rightarrow 2x = 7\\
\Rightarrow x = \dfrac{7}{2}
 $
So, the second solution of the linear equation is
 $ x = \dfrac{7}{2} $
 $ y = 0 $
In two-dimensional coordinates form $ \left( {x,y} \right) $ , the second solution is $ \left( {\dfrac{7}{2},0} \right) $ .
And finally,
Substituting $ y = 1 $ in the linear equation we have,
 $
\Rightarrow 2x + 1 = 7\\
\Rightarrow 2x = 6\\
\Rightarrow x = \dfrac{6}{2}\\
\Rightarrow x = 3
 $
So, the third solution of the linear equation is
 $ x = 3 $
 $ y = 1 $
In two-dimensional coordinates form $ \left( {x,y} \right) $ , the third solution is $ \left( {3,1} \right) $ .
Therefore, the three solutions of the equation $ 2x + y = 7 $ are $ \left( {0,7} \right) $ , $ \left( {\dfrac{7}{2},0} \right) $ and $ \left( {3,1} \right) $ .
The correct answer is –
(A) $ \left( {0,7} \right),\left( {\dfrac{7}{2},0} \right){\rm{ and }}\left( {3,1} \right) $
So, the correct answer is “Option A”.

Note: It should be noted that the value of the arbitrary constants could be any number but, in this case, we are considering only those values which will yield our answer. For example, the value of the variable $ y = 1 $ is substituted in the linear equation because all the options given in the question had the value of $ y $ coordinate as 1 and substituting only this value of $ y $ will yield our answer.