A pair of linear equations which has a unique solution $x = 2, y = -3$ is:
$x + y = -1$ and $2x - 3y = -5$
$2x + 5y = -11$ and $4x + 10y = -22$
$2x - y = 7$ and $3x + 2y = 0$
$x - 4y = 14$ and $5x - y = 13$
Step-by-Step Solution
Key Concept: Check which system is satisfied by $x=2, y=-3$ and has $\dfrac{a_1}{a_2} <br>eq \dfrac{b_1}{b_2}$.
Stepwise Solution:
Test option 3: $2(2) - (-3) = 4 + 3 = 7$ ✓ and $3(2) + 2(-3) = 6 - 6 = 0$ ✓. [0.5 Mark]
Ratio check: $\dfrac{a_1}{a_2} = \dfrac{2}{3}$ and $\dfrac{b_1}{b_2} = -\dfrac{1}{2}$. Since $\dfrac{2}{3}
eq -\dfrac{1}{2}$, it has a unique solution $(2,-3)$. [0.5 Mark]
Marking Scheme:
• Checking substitution of $(2,-3)$: 0.5 Mark
• Verifying unique solution ratio condition: 0.5 Mark
Correct Answer: $2x - y = 7$ and $3x + 2y = 0$