What is found by comparing the ratios of (a) and (b) in the equations (19x+12y=71) and (9x+6y=35)?

Correct answer: (19/9 \ne 12/6), so one unique solution

For two linear equations, the ratios of corresponding coefficients determine how their lines are related. If \(a_1/a_2\) is different from \(b_1/b_2\), the lines have different slopes and therefore intersect at exactly one point. The pair is then consistent and independent, with one unique solution. Equal first two ratios alone would not be enough to conclude infinitely many solutions; the constant-term ratio would also need to agree.

Here, \(19/9\) is not equal to \(12/6\), because \(12/6=2\), whereas \(19/9\) is about \(2.11\). Thus the coefficient ratios are unequal, so the two lines meet once and the equations have a unique solution. Choice C correctly states this conclusion. Choice A is not valid because its displayed equality is false, and choice D also uses an equality that does not hold.