The pair of linear equations $2x = 5y + 6$ and $15y = 6x - 18$ represents two lines which are :
Generated by claude-sonnet-4-6 · 2026-06-15 10:33 · grounding rag
Model Answer
Rewriting: $2x - 5y - 6 = 0$ and $6x - 15y - 18 = 0$.
$$\frac{a_1}{a_2} = \frac{2}{6} = \frac{1}{3}, \quad \frac{b_1}{b_2} = \frac{-5}{-15} = \frac{1}{3}, \quad \frac{c_1}{c_2} = \frac{-6}{-18} = \frac{1}{3}$$
Since $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$, the lines are (c) coincident.
Source: Chapter 3, Section 3.2
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Explanation
- Always rewrite both equations in standard form $ax + by + c = 0$ before comparing ratios.
- The key rule: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ → coincident (infinitely many solutions); $= \neq$ → parallel; $\neq$ → intersecting.
- A common mistake is stopping after checking only two ratios — check all three.