Question

Solve the following simultaneous linear equations
 \[97x + 53y = 177\]
 \[53x + 97y = 573\]

Answer 1

Hint: We are given with linear equations. They can be solved either by substitution method or elimination method. Here but before using any of the methods we will simplify the equations by adding them once and then subtracting them once. That pair of linear equations so obtained will help in using the approach to solve the equations very easily. We will use the elimination method here.

Complete step by step solution:
Let us first number the given equations.
 \[97x + 53y = 177\] …….I
 \[53x + 97y = 573\] …….II
Now we will add the equations
 \[97x + 53y + 53x + 97y = 177 + 573\]
Taking similar terms on one side,
 \[97x + 53x + 53y + 97y = 177 + 573\]
Adding similar terms we get
 \[150x + 150y = 750\] ……III
Now dividing by 150 we will get
 $ x+y = 5 $
Now we will subtract the equations
 \[97x + 53y - 53x - 97y = 177 - 573\]
Taking similar terms on one side,
 \[97x - 53x + 53y - 97y = 177 - 573\]
Now subtracting the terms we get,
 \[44x - 44y = - 396\]
Dividing both sides by 44 we get,
 \[x - y = - 9\] ……IV
Now we have simplified linear equations to solve,
On adding equation III and IV we get,
 \[x + y + x - y = 5 + \left( { - 9} \right)\]
Cancelling y and adding similar terms,
 \[2x = - 4\]
On dividing the term we get,
 \[x = - 2\]
To find the value of y we will use equation III
 \[x + y = 5\]
Putting the value of x in this equation we get,
 \[ - 2 + y = 5\]
Taking constant on one side we get,
 \[y = 5 + 2\]
 \[y = 7\]
This is our answer such that \[x = - 2\] and \[y = 7\] .
So, the correct answer is “ \[x = - 2\] and \[y = 7\] ”.

Note: Pair of linear equations has two equations with two variables. In order to solve them we either can use substitution method or can use elimination method. Here in the example above we have used elimination methods.
In the elimination method we make the coefficient of one of the variables same. Then perform necessary mathematical operations like addition or subtraction in between the equation so that the same coefficient is deleted from the pair providing the value of another variable whereas in the substitution method we take one of the equations and convert it in the form of another variable. Then putting that value in the equation we find the previous variable.