The linear equations \(3x+5y=19\) and \(10x-3y=24\) have the solution \(x=\frac{\alpha}{3}\) and \(y=\frac{\beta}{2}\). Then the value of \(\alpha+\beta\) is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 13
Solve the two equations for \(x\) and \(y\) by elimination, then read off \(\alpha\) and \(\beta\).
Making the \(y\) terms equal and opposite:
\(3\times(3x+5y=19)\): \(9x+15y=57\) (1)
\(5\times(10x-3y=24)\): \(50x-15y=120\) (2)
Adding (1) and (2), the \(y\) terms cancel: \(59x=177\), so \(x=3\)
Putting \(x=3\) in the first equation: \(9+5y=19\), so \(y=2\)
Finding \(\alpha\) and \(\beta\):
From \(x=\frac{\alpha}{3}=3\): \(\alpha=9\)
From \(y=\frac{\beta}{2}=2\): \(\beta=4\)
Their sum: \(\alpha+\beta=9+4=13\)