Maths · Test of consistency and solution of simultaneous linear equations in two or three variables using matrices
If the pair of equations possess
If \( 2 a=b, \) the pair of equations \( a x+ \) \( b y=2 a^{2}-3 b^{2}, x+2 y=2 a-6 b \) possess
- A. no solution
- B. only one solutions
- C. only two solutions
- D. an infinite number of solutions
Step-by-step solution
Given 2a = b, so b = 2a. Substitute into the equations: first becomes a x + 2a y = 2a^2 - 3(4a^2) = -10a^2, i.e., a(x+2y) = -10a^2; second becomes x+2y = 2a - 6(2a) = -10a. If a ≠ 0, divide the first by a to get x+2y = -10a, identical to the second, so infinitely many solutions. If a = 0, then b=0, first equation becomes 0=0, second is x+2y=0, also infinitely many solutions. Thus the system always has infinitely many solutions.
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