10. A pair of linear equations which has a unique solution \(x = 2\), \(y = -3\) is
Step 1: Check option (A).
Equation 1: \(x + y = -1\). Substituting \(x = 2, y = -3\):
\(2 + (-3) = -1\) ✓ satisfied.
Equation 2: \(2x - 3y = -5\). Substituting:
\(2(2) - 3(-3) = 4 + 9 = 13 \ne -5\).
So, option (A) is not correct.
Step 2: Check option (B).
Equation 1: \(2x + 5y = -11\). Substituting:
\(2(2) + 5(-3) = 4 - 15 = -11\) ✓ satisfied.
Equation 2: \(4x + 10y = -22\). Substituting:
\(4(2) + 10(-3) = 8 - 30 = -22\) ✓ satisfied.
But notice: the second equation is exactly 2 × (first equation). So both equations represent the same line ⇒ infinitely many solutions, not a unique one.
Step 3: Check option (C).
Equation 1: \(2x - y = 1\). Substituting:
\(2(2) - (-3) = 4 + 3 = 7 \ne 1\).
So, option (C) is not correct.
Step 4: Check option (D).
Equation 1: \(x - 4y - 14 = 0\). Substituting:
\(2 - 4(-3) - 14 = 2 + 12 - 14 = 0\) ✓ satisfied.
Equation 2: \(5x - y - 13 = 0\). Substituting:
\(5(2) - (-3) - 13 = 10 + 3 - 13 = 0\) ✓ satisfied.
Both equations are satisfied and they are not multiples of each other ⇒ two distinct lines intersecting at exactly one point.
Conclusion. Option (D) gives a pair of equations with a unique solution at \((2, -3)\).