Question

Identify a solution of the linear equation \[x + y = 0\].
(a) \[\left( {2,2} \right)\]
(b) \[\left( {2,3} \right)\]
 (c) \[\left( {1, - 1} \right)\]
(d) \[\left( {4, - 4} \right)\]

Answer 1

Hint: Here, we need to identify which of the options are solutions of the linear equation \[x + y = 0\]. A solution of a linear equation is one, which satisfies the linear equation. We can say \[\left( {a,b} \right)\] is the solution of the linear equation \[x + y = 0\] only when it satisfies the equation. This means that if \[x = a\] and \[y = b\], then the expression \[x + y\] should be equal to 0. Here we will check each option one by one by substituting the values in the given linear equation.

Complete step-by-step answer:
We will substitute the coordinates given in the options one by one into the linear equation. The co-ordinates which result in \[x + y = 0\] will be the required solutions of the linear equation.
First, let us verify whether \[\left( {2,2} \right)\] is a solution of the given linear equation or not.
Substituting \[x = 2\] and \[y = 2\] in the expression \[x + y\], we get
\[\begin{array}{l}x + y = 2 + 2\\ \Rightarrow x + y = 4\end{array}\]
We can observe that the expression \[x + y\] is not equal to 0.
So, \[\left( {2,2} \right)\] is not a solution of the given equation. Hence, option (a) is incorrect.
Next, let us verify whether \[\left( {2,3} \right)\] is a solution of the given linear equation or not.
Substituting \[x = 2\] and \[y = 3\] in the expression \[x + y\], we get
\[\begin{array}{l}x + y = 2 + 3\\ \Rightarrow x + y = 5\end{array}\]
We can observe that the expression \[x + y\] is not equal to 0.
Therefore, \[\left( {2,3} \right)\] is not a solution of the given equation. Option (b) is incorrect.
Now, let us verify whether \[\left( {1, - 1} \right)\] is a solution of the given linear equation or not.
Substituting \[x = 1\] and \[y = - 1\] in the expression \[x + y\], we get
\[\begin{array}{l}x + y = 1 + \left( { - 1} \right)\\ \Rightarrow x + y = 1 - 1\\ \Rightarrow x + y = 0\end{array}\]
We can observe that the expression \[x + y\] is equal to 0.
Therefore, \[\left( {1, - 1} \right)\] is a solution of the given equation. Option (c) is correct.
Finally, let us verify whether \[\left( {4, - 4} \right)\] is a solution of the given linear equation or not.
Substituting \[x = 4\] and \[y = - 4\] in the expression \[x + y\], we get
\[\begin{array}{l}x + y = 4 + \left( { - 4} \right)\\ \Rightarrow x + y = 4 - 4\\ \Rightarrow x + y = 0\end{array}\]
We can observe that the expression \[x + y\] is equal to 0.
Therefore, \[\left( {4, - 4} \right)\] is a solution of the given equation. Option (d) is correct.
Hence, we see that there are two correct options, option (c) and option (d).

Note: We can also solve this problem by rewriting the equation and comparing it with the given points.
Rewriting the linear equation \[x + y = 0\], we get
\[y = - x\]
This means that if \[x = 2\], then \[y = - 2\].
Therefore, \[\left( {2,2} \right)\] and \[\left( {2,3} \right)\] cannot be the solution of the linear equation.
Now, substituting \[x = 1\], we get \[y = - 1\].
Therefore, \[\left( {1, - 1} \right)\] is a solution of the given equation. Option (c) is correct.
Next, substituting \[x = 4\], we get \[y = - 4\].
Therefore, \[\left( {4, - 4} \right)\] is a solution of the given equation. Option (d) is correct.