Question

If $ 331a+247b=746 $ and $ 247a+331b=410 $, then find the value of a.
(a) -1
(b) 2
(c) 3
(d) 6

Answer 1

Hint: Here, we will use the method of elimination to solve the given system of linear equations. Since, we have to find the value of ‘a’, we will eliminate ‘b’ using the given equations. We will multiply the first equation by 331 and the second equation by 247. Thereafter, we will subtract any one of the equations from the other to obtain the value of ‘a’.

Complete step-by-step answer:
We know that there are various methods of solving a linear equation like:
(1) Graphical method
(2) Substitution method
(3) Elimination method
Here, we will prefer to use the elimination method.
Since, the given equations are:
 $ 331a+247b=746.........\left( 1 \right) $
  $ 247a+331b=410........\left( 2 \right) $
On multiplying equation (1) by 331 and equation (2) by 247, we get:
  $ \begin{align}
  & 331\times \left( 331a+247b \right)=331\times 746 \\
 & \Rightarrow {{\left( 331 \right)}^{2}}a+331\times 247b=331\times 746.......\left( 3 \right) \\
\end{align} $
And,
 $ \begin{align}
  & 247\times \left( 247a+331b \right)=247\times 410 \\
 & \Rightarrow {{\left( 247 \right)}^{2}}a+247\times 331b=247\times 410.........\left( 4 \right) \\
\end{align} $
On subtracting equation (4) from equation (3), we get:
 $ \begin{align}
  & {{\left( 331 \right)}^{2}}a+331\times 247b-{{\left( 247 \right)}^{2}}a-247\times 331b=\left( 331\times 746 \right)-\left( 247\times 410 \right) \\
 & \Rightarrow a\left( {{331}^{2}}-{{247}^{2}} \right)=145656 \\
\end{align} $
We know the identity that:
 $ \left( {{x}^{2}}-{{y}^{2}} \right)=\left( x-y \right)+\left( x+y \right) $
So, on applying this identity on above equation, we get:
 $ \begin{align}
  & a\left( 331-247 \right)\left( 331+247 \right)=145656 \\
 & \Rightarrow a\left( 84\times 578 \right)=145656 \\
 & \Rightarrow a=\dfrac{145656}{48552} \\
 & \Rightarrow a=3 \\
\end{align} $
So, the value of ‘a’ comes out to be 3.
Hence, option (c) is the correct answer.

Note: Students must note here that while using the elimination method any one of the two variables, that is, a or b can be eliminated. But we have to find the value of ‘a’ so we eliminate it by using the given equations. We can also eliminate ‘a’ and find the value of ‘b’ and then putting the value of ‘b’ in any one of the equations, we can easily get the value of ‘a’. We can also use here the method of substitution in which we have to just write ‘a’ in terms of ‘b’ using one of the equations and then we have to put this in the other equation. Thereafter, we will have an equation in ‘a’ only and by solving this we can get the value of ‘a’.