Sample Question 3: Is the pair of equations \(x + 2y - 3 = 0\) and \(3x + 6y - 9 = 0\) consistent? Justify your answer.
Yes. They are dependent (coincident) and hence consistent.
Compute ratios.
\(\dfrac{a_1}{a_2} = \dfrac{1}{3}\)
\(\dfrac{b_1}{b_2} = \dfrac{2}{6} = \dfrac{1}{3}\)
\(\dfrac{c_1}{c_2} = \dfrac{-3}{-9} = \dfrac{1}{3}\)
All three ratios are equal, so the two equations represent the same line.
Therefore, they are coincident with infinitely many solutions, i.e., consistent.