Question

Solve the following systems of equations:
99x + 101y = 499
101x + 99y = 501

Answer 1

Hint – In this question simply add the two given equations and then divide by 200 all throughout, then we will be getting the value of x in terms of y. Substitute it back to any one of the equations given to get the values.

Complete step-by-step answer:
Given system of equations:
$99x + 101y = 499$............................. (1)
$101x + 99y = 501$..................... (2)
Now add these two equations we have,
$ \Rightarrow 99x + 101y + 101x + 99y = 499 + 501$
$ \Rightarrow 200x + 200y = 1000$
Now divide by 200 throughout we have,
$ \Rightarrow x + y = 5$
$ \Rightarrow x = 5 - y$......................... (3)
Now substitute this value in equation (1) we have,
$ \Rightarrow 99\left( {5 - y} \right) + 101y = 499$
Now simplify this equation we have,
$ \Rightarrow 99 \times 5 - 99y + 101y = 499$
$ \Rightarrow 2y = 499 - 495 = 4$
Now divide by 2 we have,
$ \Rightarrow y = \dfrac{4}{2} = 2$
Now from equation (3) we have,
$ \Rightarrow x = 5 - 2 = 3$
So the required solution of the given system of equation is
$ \Rightarrow \left( {x,y} \right) = \left( {3,2} \right)$
So this is the required answer.

Note – This question could have been solved by another method in this simply we would be using a substitution method, using the first relation take out x in terms of y and put this relation back into the second equation, to get the value of the unknown variable. This method would be lengthy, therefore before applying this the equations are simplified by addition of them.