What conclusion follows from comparing the ratios of a and b in 7x + 10y = 39 and 5x + 8y = 31?

7/5 ≠ 10/8, so there is one unique solution

For the pair 7x + 10y = 39 and 5x + 8y = 31, compare the ratios of the corresponding coefficients. The x-coefficient ratio is 7/5 and the y-coefficient ratio is 10/8 = 5/4. These are not equal: cross multiplication gives 7 × 8 = 56, while 10 × 5 = 50. Therefore the two lines have different slopes and must intersect at exactly one point. The pair is consequently consistent and independent, with one unique solution. Comparing the constant ratio is unnecessary once the first two ratios differ. No solution would require equal x- and y-coefficient ratios but a different constant ratio; infinitely many solutions would require all three ratios to be equal. Thus option C states the correct conclusion.

This is a Class 10 Mathematics question. Chapter: Pair of Linear Equations in Two Variables. Topic: Conditions for solvability.