Question

Solve for x and y:
 $ 99x+101y=499 $
 $ 101x+99y=501 $

Answer 1

Hint: A pair of simultaneous linear equations in two variables can be solved by either Cramer's rule, by elimination, or by substitution.
In order to eliminate one of the variables, make its coefficients equal in both the equations by multiplying / dividing the equations by some number, and then add / subtract the resulting equations.
In order to avoid the lengthy multiplications, add and subtract both the equations to get two new equations with smaller coefficients.

Complete step by step answer:
The given pair of simultaneous equations is:
 $ 99x+101y=499 $ ... (1)
 $ 101x+99y=501 $ ... (2)
On adding the equations (1) and (2), we get:
 $ 200x+200y=1000 $
Dividing all the terms by 200, gives us:
⇒ $ x+y=5 $ ... (3)
On subtracting equation (1) from equation (2), we get:
 $ 2x-2y=2 $
Dividing all the terms by 2, gives us:
⇒ $ x-y=1 $ ... (4)
Since the coefficient of y are equal and of opposite sign, adding equations (3) and (4) will give us:
 $ 2x=6 $
⇒ $ x=3 $
And, substituting this value of x in equation (3) [or equation (4)] gives us:
 $ 3+y=5 $
⇒ $ y=2 $
∴ The solution of the given system of equations is $ x=3 $ and $ y=2 $ .
Check:
 $ 99\times 3+101\times 2 $
= $ (100-1)\times 3+(100+1)\times 2 $
= $ 300-3+200+2 $
= 499
And,
 $ 101\times 3+99\times 2 $
= $ (100+1)\times 3+(100-1)\times 2 $
= $ 300+3+200-2 $
= 501

Note: The solutions to the equations $ {{a}_{1}}x+{{b}_{1}}y={{c}_{1}} $ and $ {{a}_{2}}x+{{b}_{2}}y={{c}_{2}} $ can be determined using Cramer's rule:
 $ D=\left| \begin{matrix} {{a}_{1}} & {{a}_{2}} \\ {{b}_{1}} & {{b}_{2}} \\ \end{matrix} \right| $ , $ {{D}_{x}}=\left| \begin{matrix} {{c}_{1}} & {{c}_{2}} \\ {{b}_{1}} & {{b}_{2}} \\ \end{matrix} \right| $ , $ {{D}_{y}}=\left| \begin{matrix} {{a}_{1}} & {{a}_{2}} \\ {{c}_{1}} & {{c}_{2}} \\ \end{matrix} \right| $ .
The values of the variables x and y are given by: $ x=\dfrac{{{D}_{x}}}{D} $ and $ y=\dfrac{{{D}_{y}}}{D} $ .
If, D = 0, the system is:
Either consistent and has infinitely many solutions. In this case $ {{D}_{x}}={{D}_{y}}=0 $ .
Or it is inconsistent, i.e. it has no solutions. In this case $ {{D}_{x}}\ne 0 $ and $ {{D}_{y}}\ne 0 $ .
Consistent equations with infinitely many solutions are also known as dependent equations, because both are multiples of the same equation.