The pair of linear equations \(kx+2y=5\) and \(3x+y=1\) has a unique solution if:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(k\neq 6\)
Two linear equations \(a_1x+b_1y=c_1\) and \(a_2x+b_2y=c_2\) have a unique solution when their lines intersect, that is, when \(\frac{a_1}{a_2}\neq\frac{b_1}{b_2}\). If the two ratios are equal, the lines have the same slope and are either parallel (no solution) or the same line (infinitely many solutions).
Ratio of the \(x\)-coefficients: \(\frac{a_1}{a_2}=\frac{k}{3}\)
Ratio of the \(y\)-coefficients: \(\frac{b_1}{b_2}=\frac{2}{1}=2\)
The ratios must differ: \(\frac{k}{3}\neq2\), that is, \(k\neq6\)