Question

Solve the following pair of equations.
 \[25x - 24y = 197\] and \[24x - 25y = 195\]?
A). \[x = 1,{\text{ }}y = - 3\]
B). \[x = 9,{\text{ }}y = - 3\]
C). \[x = 5,{\text{ }}y = - 3\]
D). \[x = 3,{\text{ }}y = - 3\]

Answer 1

Hint: Here in this question, given the system of linear equations. We have to find the unknown values that are \[X\] and \[Y\] solving these equations by using the elimination method. In the elimination method either we add or subtract the equations to find the unknown values of\[X\] and \[Y\].

Complete step-by-step solution:
Let us consider the equation and we will name it as (1) and (2)
\[25x - 24y = 197\]----------(1)
\[24x - 25y = 195\]----------(2)
If we apply the elimination method it will be difficult to solve.
So we reduce the equation.
First add equation (1) and equation (2) we have
\[25x - 24y + 24x - 25y = 195 + 197\]
\[49x - 49y = 392 - - - - (3)\]
Now subtract equation (1) and (2) we have,
\[25x - 24y - \left( {24x - 25y} \right) = 197 - 195\]
\[25x - 24y - 24x + 25y = 2\]
\[x + y = 2 - - - - (4)\].
Thus we have,
\[49x - 49y = 392 - - - - (3)\]
\[x + y = 2 - - - - (4)\]
Now we have to solve these two equations to find the unknown
Multiply (3) by 1 and Multiply (4) by 49, then we get
\[49x - 49y = 392\]
\[49x + 49y = 98\]
Now adding we have
\[ 49x - 49y = 392 \\
  49x + 49y = 98 \\
 \overline{98x - 0 = 490} \]
\[\Rightarrow 98x = 490\]
Divide 98 on both sides, then
\[\therefore \,\,x = 5\,\]
Now to find the value of ‘y’ put the value of ‘x’ in any one of the equations. That is Equation (1), (2) (3) and (4).
To simplify easily we put \[\,x = 5\,\]in equation (4).
\[x + y = 2\]
\[\Rightarrow 5 + y = 2\]
\[\Rightarrow y = 2 - 5\]
\[\Rightarrow y = - 3\]
Thus, we have \[x = 5\,\] and \[y = - 3\].
Hence the required option is (c).

Note: In this type of question while eliminating the term we must be aware of the sign where we change the sign by the alternate sign. In this, we have not altered because without altering we can cancel the ‘y’ term. In this, we have a chance to verify our answers. In the elimination method, we have made the one term have the same coefficient such that it will be easy to solve the equation.