Question

Solve the following pairs of equations: $41x+53y=135;53x+41y=147$.
A. $x=2;y=1$
B. $x=1;y=4$
C. $x=2;y=7$
D. $x=3;y=-4$

Answer 1

Hint: To solve this question, we will first add both the given equations and then we will subtract any one of the equations from the other. By applying these operations, we will get a pair of equations and then we will apply the elimination method to get the value of x and y.

Complete step-by-step solution -
In this question, we have been given a pair of equations, that is, $41x+53y=135;53x+41y=147$, and we have been asked to find the value of x and y. So, to solve this we will consider,
$41x+53y=135\ldots \ldots \ldots \left( i \right)$ and $53x+41y=147\ldots \ldots \ldots \left( ii \right)$
Now, we will add equation (i) and (ii). So, we will get,
$41x+53x+53y+41y=135+147$
And, we know that the like terms can show arithmetic operation. Therefore, we can write the above equation as follows,
$94x+94y=282$
Here, we can take 94 as common, so we get the equation as,
$\begin{align}
  & 94\left( x+y \right)=282 \\
 & \Rightarrow x+y=\dfrac{282}{94} \\
 & \Rightarrow x+y=3\ldots \ldots \ldots \left( iii \right) \\
\end{align}$
Now, we will subtract equation (i) from equation (ii). So, we will get the following equation,
$53x-41x+41y-53y=147-135$
And, we know that we can subtract the like terms from each other. So, we get the above equation as,
$12x-12y=12$
Here, we will take 12 as common. So, the equation will become,
$\begin{align}
  & 12\left( x-y \right)=12 \\
 & \Rightarrow x-y=\dfrac{12}{12} \\
 & \Rightarrow x-y=1\ldots \ldots \ldots \left( iv \right) \\
\end{align}$
Now, we have obtained two new and easier equations. So, to get the value of x and y, we will apply the elimination method to equations (iii) and (iv). So, we will add equations (iii) and (iv). Therefore, we get,
$\begin{align}
  & x+x+y-y=3+1 \\
 & \Rightarrow 2x+0=4 \\
 & \Rightarrow x=\dfrac{4}{2} \\
 & \Rightarrow x=2\ldots \ldots \ldots \left( v \right) \\
\end{align}$
Now, we will put the value of x = 2 in equation (iii). So, we will get,
$\begin{align}
  & 2+y=3 \\
 & \Rightarrow y=3-2 \\
 & \Rightarrow y=1\ldots \ldots \ldots \left( vi \right) \\
\end{align}$
From equations (v) and (vi), we get x = 2; y = 1.
Hence, option A is the correct option.

Note: While solving this question, one can think of applying the elimination or substitution method at the very first step. It is not wrong, but not very correct too as it would make the question very complicated and lengthy. So, to reduce the complicacy, we will first add both the equations, then subtract each other and apply the elimination or substitution method to the new equations thus obtained.