2. The pair of equations \(x + 2y + 5 = 0\) and \(-3x - 6y + 1 = 0\) have
a unique solution
exactly two solutions
infinitely many solutions
no solution
Step 1: Write coefficients.
From the first equation \(x + 2y + 5 = 0\):
\(a_1 = 1,\; b_1 = 2,\; c_1 = 5\).
From the second equation \(-3x - 6y + 1 = 0\):
\(a_2 = -3,\; b_2 = -6,\; c_2 = 1\).
Step 2: Check ratios of coefficients.
\(\dfrac{a_1}{a_2} = \dfrac{1}{-3} = -\dfrac{1}{3}\).
\(\dfrac{b_1}{b_2} = \dfrac{2}{-6} = -\dfrac{1}{3}\).
So, \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2}\).
Step 3: Check constant ratio.
\(\dfrac{c_1}{c_2} = \dfrac{5}{1} = 5\).
This is not equal to \(-\dfrac{1}{3}\).
Step 4: Interpret.
Since the first two ratios are equal but different from the third, the lines are parallel and distinct.
Hence, the pair of equations has no solution.