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Question
Solve each of the following systems of equations by the method of cross-multiplication
x + ay = b
ax − by = c
Solution
The given system of equations may be written as
x + ay - b = 0
ax − by - c = 0
Here,
`a_1 = 1, b_1 = a, c_1 = -b`
`a_2 = a, b_2 = -b, c_2 = -c`
By cross multiplication, we get
`=> x/((a)xx(-c)-(-b)xx(-b)) = (-y)/(1xx(-c)-(-b)xxa) = 1/(1xx(-b)-axxa)`
`=> x/(-ac-b^2) = (-y)/(-c + ab) = 1/(-b-a^2)`
Now
`x/(-ac -b^2) = 1/(-b-a^2)`
`=> x =- (-ac - b^2)/(-b-a^2)`
`= (b^2 + ac)/(a^2 + b)`
And
`(-y)/(-c + ab) = 1/(-b-a^2)`
`=> -y = (ab -c)/(-(a^2 +b))`
`=> y = (ab -c)/(a^2 + b)`
Hence `x = (ac + b^2)/(a^2 + b), y = (ab -c)/(a^2 + b)` is the solution of the given system of the equations.
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