Which condition is correct for the equations (10x+py=50) and (2x+7y=19) to have a unique solution?

Correct answer: \(p\ne35\)

For two linear equations to have a unique solution, the ratios of the corresponding coefficients must be unequal: \(\frac{10}{2}\ne\frac{p}{7}\). Thus, \(5\ne\frac{p}{7}\), which gives \(p\ne35\). If \(p=35\), the ratios of the coefficients of \(x\) and \(y\) become equal; since \(\frac{50}{19}\ne\frac{10}{2}\), the equations then have no solution rather than a unique solution. Exam tip: for a unique solution, check \(\frac{a_1}{a_2}\ne\frac{b_1}{b_2}\).