Question

Solve the following equations:
$x+y=a+b$ and $ax-by={{a}^{2}}-{{b}^{2}}$

Answer 1

Hint: We solve the two linear equations by using the elimination method, where we try to eliminate one of the variables by making the coefficients of that variable same in the two equations.

Complete step-by-step answer:
In the given set of equations we will use the elimination method.
Here the coefficient of any one variable is made equal and of opposite sign to the coefficient of the same variable of the second equation.
Then add the equations to eliminate that variable in order to determine the value of the other variable.

Given equations are
$x+y=a+b$……………………………….(i)
$ax-by={{a}^{2}}-{{b}^{2}}$………………………………(ii)

In these equations no variable has the same coefficient. Hence, to make same coefficient we need to multiply equation (i) with b
$\Rightarrow b\left( x+y \right)=b\left( a+b \right)$
$\Rightarrow bx+by=ab+{{b}^{2}}$ ……………………………………(iii)

Adding equation (iii) and equation (ii) we get,
$\Rightarrow ax-by+bx+by={{a}^{2}}-{{b}^{2}}+ab+{{b}^{2}}$
$\Rightarrow ax+bx={{a}^{2}}+ab$
$\Rightarrow \left( a+b \right)x=a\left( a+b \right)$

On dividing both side by $\left( a+b \right)$
$\Rightarrow \dfrac{\left( a+b \right)x}{(a+b)}=\dfrac{a\left( a+b \right)}{\left( a+b \right)}$
$\Rightarrow x=a$
On substituting the value of x in equation (i), we get,
$\Rightarrow a+y=a+b$
$\Rightarrow y=b$
Thus, complete solutions for this equation can be written as
$x=a,\,y=b$

Note: If the variable we are trying to eliminate has the same sign in both the equations, then one of the equations should be subtracted from the other one to eliminate the variable. This problem can also be solved by using the substitution method.