Question
Solve each of the following systems of equations by the method of cross-multiplication :
mx – my = m2 + n2
x + y = 2m
Solution
The given system of equations may be written as
`mx - ny - (m^2 + n^2) = 0`
`x + y - 2m = 0
Here
`a_1 = m, b_1 = -n, c_1 = -(n^2 + n^2)`
`a_2 = 1, b_2 = 1, c_2 = -2m`
By cross multiplication, we have
`x/(2mn + (m^2 + n^2)) = (-y)/(-2m^2 + (m^2 + n^2)) = 1/(m + n)`
`=> x/(2mn + m^2 + n^2) = (-y)/(-m^2 + n^2) = 1/(m + n)`
`=> x/(m + n)^2 = (-y)/(-m^2 + n^2) = 1/(m + n)`
Now
`x/(m + n)^2 = 1/(m + n)`
`=> x = (m + n^2)/(m + n)`
=> x = m + n
And
`(-y)/(-m^2 + n^2) = 1/(m + n)`
`=> -y = (-m^2 + n^2)/(n + n)`
`=> y = (m^2 - n^2)/(m + n)`
`=> y = ((m - n)(m + n))/(m + n)`
=> y = m - n
Hence, x = m + n, y = m - n is the solution of the given system of equation.
