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प्रश्न
Solve each of the following systems of equations by the method of cross-multiplication
ax + by = a − b
bx − ay = a + b
उत्तर
The given system of equations is
ax + by = a − b .....(i)
bx − ay = a + b ....(ii)
here
`a_1 - a, b_1 = b, c_1 = b - a`
`a_2 = b, b_2 = -a, c_2 = -a-b`
By cross multiplication, we get
`=> x/((-a-b)xx(b)-(b-a)xx(-a)) = (-y)/((-a-b)xx(a)-(b-a)xx(-b)) = 1/(-axxa-bxxb)`
`=> x/(-ab -b^2 + ab -a^2) = (-y)/(-a^2-ab-b^2 + ab)= 1/(-a^2 - b^2)`
`=> x/(-b^2 - a^2) = (-y)/(=a^2 - b^2) = 1/(-a^2 - b^2)`
Now
`x/(-b^2-a^2) = 1/(-a^2-b^2)`
`=> x = (-b^2 - a^2)/(-a^2 - b^2)`
`=(-(b^2 + a^2))/(a^2 + b^2)`
`= (a^2 + b^2)/(a^2 + b^2)`
=> x= 1
`(-y)/(-a^2 -b^2) = 1/(-a^2-b^2)`
`=> -y = (-(a^2 + b^2))/(-(a^2 + b^2))`
=> -y = 1
=> y = -1
Hence x = 1, y = -1 is the solution of the given system of the equations
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