Question

Find the values of \[x,y\] if \[\left( x+3,5 \right)=\left( 6,2x+y \right)\]
(a) \[x=3,y=-1\]
(b) \[x=6,y=2\]
(c) \[x=2,y=3\]
(d) None of the above.

Answer 1

Hint: We solve this problem by using the simple definition of equal points.
If two points in a co – ordinate plane are said to be equal if and only if their corresponding co – ordinates are equal that is if the two points \[A\left( x,y \right),B\left( p,q \right)\] are said to be equal if and only if
\[\begin{align}
  & \Rightarrow x=p \\
 & \Rightarrow y=q \\
\end{align}\]
By using this formula we calculate the required values.

Complete step-by-step answer:
We are given that \[\left( x+3,5 \right)=\left( 6,2x+y \right)\]
We know that if two points in a co – ordinate plane are said to be equal if and only if their corresponding co – ordinates are equal that is if the two points \[A\left( x,y \right),B\left( p,q \right)\] are said to be equal if and only if
\[\begin{align}
  & \Rightarrow x=p \\
 & \Rightarrow y=q \\
\end{align}\]
By using the above formula let us take the first co – ordinate in the given two points then we get
\[\Rightarrow x+3=6\]
Now by subtracting with 3 on both sides we get
\[\begin{align}
  & \Rightarrow x+3-3=6-3 \\
 & \Rightarrow x=3 \\
\end{align}\]
Therefore the value of \['x'\] is 3
Now let us take the second co – ordinate from the given two points then we get
\[\Rightarrow 5=2x+y\]
Now, by substituting the value that is \[x=3\] in above equation we get
\[\Rightarrow y+2\left( 3 \right)=5\]
Now by subtracting with 6 on both sides we get
\[\begin{align}
  & \Rightarrow y+6-6=5-6 \\
 & \Rightarrow y=-1 \\
\end{align}\]
Therefore, the value of \['y'\] is -1
So, we can conclude that the values of \[x,y\] are \[x=3,y=-1\]

So, the correct answer is “Option A”.

Note: We can solve this problem in another method that is a substitution method.
We are given that the equation as
\[\left( x+3,5 \right)=\left( 6,2x+y \right)\]
Now, let us substitute the option (a) that is \[x=3,y=-1\] in above equation then we get
\[\begin{align}
  & \Rightarrow \left( \left( 3+3 \right),5 \right)=\left( 6,\left( 2\left( 3 \right)-1 \right) \right) \\
 & \Rightarrow \left( 6,5 \right)=\left( 6,5 \right) \\
\end{align}\]
Here, we can see that both LHS and RHS are equal to each other.
So, we can conclude that the values of \[x,y\] are \[x=3,y=-1\]
Therefore, option (a) is the correct answer.