Question

Solve the following pair of equation
$41x+53y=135$ , $53x+41y=147$
A. $x=-4$ ; $y=2$ .
B. $x=2$ ; $y=1$ .
C. $x=\dfrac{2}{5}$ ; $y=3$ .
D. $x=-1$ ; $y=1$ .

Answer 1

Hint: In this problem we need to solve the given pair of equations. For this we will first observe the coefficients of the variables in the given equations. Now we will multiply equation one with a constant and equation two with another constant in order to make the coefficient of any one of the variables the same. Now we will subtract both the equations, then we will get the whole equation in terms of only one variable. Now we will use basic mathematical operations to get the value of one variable. After having the value of one variable we will use any one of the equations to get the value of the remaining variable.

Complete step-by-step solution:
Given equations are $41x+53y=135$ , $53x+41y=147$ .
We have only the values $41$ , $53$ as coefficients in both the given equations.
So multiplying the first equation with $53$, then we will get
$\begin{align}
  & 53\left( 41x+53y \right)=53\left( 135 \right) \\
 & \Rightarrow 2173x+2809y=7155....\left( \text{i} \right) \\
\end{align}$
Now multiplying the second given equation with $41$, then we will have
$\begin{align}
  & 41\left( 53x+41y \right)=41\left( 147 \right) \\
 & \Rightarrow 2173x+1681y=6027.....\left( \text{ii} \right) \\
\end{align}$
Now subtracting the equation $\left( \text{ii} \right)$ from the equation $\left( \text{i} \right)$ , then we will get
$\begin{align}
  & 2173x+2809y-\left( 2173x+1681y \right)=7155-6027 \\
 & \Rightarrow 2173x+2809y-2173x-1681y=1128 \\
 & \Rightarrow 1128y=1128 \\
\end{align}$
Dividing the above equation with $1128$ on both sides of the equation, then we will have
$\begin{align}
  & \dfrac{1128y}{1128}=\dfrac{1128}{1128} \\
 & \Rightarrow y=1 \\
\end{align}$
Now substituting the value of $y$ in the equation $41x+53y=135$, then we will get
$\begin{align}
  & 41x+53\left( 1 \right)=135 \\
 & \Rightarrow 41x=135-53 \\
 & \Rightarrow 41x=82 \\
\end{align}$
Dividing the above equation with $41$ on both sides of the above equation, then we will have
$\begin{align}
  & \dfrac{41x}{41}=\dfrac{82}{41} \\
 & \Rightarrow x=2 \\
\end{align}$
So the solution of the given pair of systems $41x+53y=135$ , $53x+41y=147$ is $x=2$ and $y=1$ .
Hence option – B is the correct answer.

Note: There are lots of methods to solve the pair of equations like matrix method, graphical method. If we plot the graph of the given equation to solve them then we will get the graph as
There are lots of methods to solve the pair of equations like matrix method, graphical method. If we plot the graph of the given equation to solve them then we will get the graph as

In the above graph also we can observe that the solution of the given pair of equations is $x=2$ and $y=1$.