Question

If \[99x + 101y = 400\] and \[101x + 99y = 600\] then \[x + y\] is…….?

Answer 1

Hint: See here all numbers are near to hundred. Even right hand sides are multiples of a hundred.
 In both equations the same coefficients are there only shuffled.

Complete step-by-step answer:
Let
\[99x + 101y = 400\] ….equation 1
\[101x + 99y = 600\] ….equation 2
We will split the terms in the form of a hundred.
\[100x - x + 100y + y = 400\] …..equation 1.1
\[100x + x + 100y - y = 600\] …..equation 2.1
First we will get \[x - y\].
Equation 2.1- equation 1.1
\[100x + x + 100y - y - (100x - x + 100y + y) = 600 - 400\]
\[100x + x + 100y - y - 100x + x - 100y - y = 200\] Multiply with minus sign to bracket.
$\Rightarrow$ $x + x - y - y = 200$
$\Rightarrow$ $2x - 2y = 200$ divide both sides by 2.
$\Rightarrow$ $x - y = 100$
Now in equation 2.1 we observe x-y. so we will use this value in equation 2.1
\[100x + x + 100y - y = 600\]
\[100x + 100y + x - y = 600\] rearranging the terms
\[
\Rightarrow 100x + 100y + 100 = 600 \\
\Rightarrow 100x + 100y = 600 - 100 \\
\Rightarrow 100x + 100y = 500 \\
\]
\[x + y = 5\] dividing both sides by hundred.

Note: This problem is having two variables and two equations. We can solve it by the usual method of finding the value of one variable and then putting it in any one equation and finding the value of the other.
That is not at all wrong but it will be time consuming here. Because if you observe,
\[
  101 = 100 + 1 \\
  99 = 100 - 1 \\
\]
And right hand side values are already in multiples of 100.
This is the first step to proceed with.
Second important note is equation 1.1- equation 2.1 will lead to the same value \[x - y = 100\].
Always remember left hand sides and right hand sides of equations are indicators to lead for the solution.