Solve the system of equations \$6(ax+by)=3a+2b\$ $6(bx-ay)=3b-2a$
- $\cfrac{1}{3}, \cfrac{1}{2}$
- $\cfrac{1}{6}, \cfrac{2}{3}$
- $\cfrac{-1}{3},\cfrac{1}{6}$
- $\cfrac{1}{2}, \cfrac{1}{3}$
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Correct answer
Explanation
Equations: 6ax + 6by = 3a + 2b and 6bx - 6ay = 3b - 2a. Multiply first by a and second by b: 6a^2x + 6aby = 3a^2 + 2ab; 6b^2x - 6aby = 3b^2 - 2ab. Adding: 6x(a^2 + b^2) = 3(a^2 + b^2) => x = 1/2. Substituting x = 1/2: 3a + 6by = 3a + 2b => 6by = 2b => y = 1/3.
AI explanation
Divide both equations by 6 to get ax + by = (3a + 2b) divided by 6 and bx - ay = (3b - 2a) divided by 6. Multiply the first equation by a and the second by b, then add them to eliminate y, resulting in (a squared + b squared)x = a. This gives x = one half. Substituting x back into the first simplified equation yields y = one third. The solution is x = one half and y = one third.