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Maths · Test of consistency and solution of simultaneous linear equations in two or three variables using matrices

Solve the following pair of equations:

Solve the following pair of equations: \( \frac{a}{x}-\frac{b}{y}=0 \) \( \frac{a b^{2}}{x}+\frac{a^{2} b}{y}=a^{2}+b^{2} \)

  • A. \( x=a b \) and \( y=b \)
  • B. \( x=a \) and \( y=b \)
  • C. \( x=b \) and \( y=a \)
  • D. \( x=b-a \) and \( y=a b \)

Step-by-step solution

From the first equation, a/x = b/y => ay = bx => y = bx/a. Substitute into the second equation: ab^2/x + a^2b/(bx/a) = a^2+b^2 => ab^2/x + a^3/x = a^2+b^2 => (ab^2+a^3)/x = a^2+b^2 => a(a^2+b^2)/x = a^2+b^2 => a/x = 1 => x = a. Then y = b*a/a = b.
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