Half of the difference between two numbers is 2. The sum of the greater number and twice the smaller number is 13. Find the numbers.
Generated by claude-sonnet-4-6 · 2026-06-15 10:35 · grounding rag
Model Answer
Let the greater number be $x$ and the smaller number be $y$.
Forming equations:
Half the difference of two numbers is 2:
$$\frac{x - y}{2} = 2 \implies x - y = 4 \tag{1}$$
Sum of greater number and twice the smaller is 13:
$$x + 2y = 13 \tag{2}$$
Solving by substitution:
From (1): $x = y + 4$
Substituting in (2):
$$(y + 4) + 2y = 13$$
$$3y = 9 \implies y = 3$$
Substituting back: $x = 3 + 4 = 7$
The two numbers are 7 and 3.
Verification: $\frac{7-3}{2} = 2$ ✓ and $7 + 2(3) = 13$ ✓
Source: Chapter 3, Section 3.3.1 Substitution Method
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Explanation
- Examiners award 1 mark for correctly forming both equations, 1 mark for solving, and 1 mark for the correct answer/verification.
- Clearly define variables at the start — this is expected in board answers.
- "Half the difference" means $\frac{x-y}{2}$, not $\frac{x}{2} - y$; be careful with phrasing.
- Always verify your answer — it shows completeness and can earn the final mark.