<p>Graphically, the pair of equations 6x - 3y + 10 = 0, 2x - y + 9 = 0 represents two lines which are </p>
<p style="display:inline">coincident</p>
<p style="display:inline">Intersect at two points</p>
<p style="display:inline">parallel</p>
<p style="display:inline">intersect at a point</p>
Step-by-Step Solution
Key Concept: The geometric relationship between two lines is determined by comparing the ratios of their x-coefficients, y-coefficients, and constant terms.
<p>Given: a<sub>1</sub> = 6, a<sub>2</sub> = 2, b<sub>1</sub> = -3, b<sub>2</sub> = -1, c<sub>1</sub> = 10 and c<sub>2</sub> =9<br />
<span class="math-tex">${a_1} = 6,{a_2} = 2,{b_1} = - 3,{b_2} = - 1,{c_1} = 10$</span> and <span class="math-tex">${c_2} = 9$</span><br />
Here <span class="math-tex">$\frac{{{a_1}}}{{{a_2}}} = \frac{6}{2} = \frac{3}{1},\frac{{{b_1}}}{{{b_2}}} = \frac{{ - 3}}{{ - 1}} = \frac{3}{1},\frac{{{c_1}}}{{{c_2}}} = \frac{{10}}{9}$</span></p>
<p>but <span class="math-tex">$\frac{c_1}{c_2}=\frac{10}{9}$</span><br />
<span class="math-tex">$\because $</span><span class="math-tex">$\frac{{{a_1}}}{{{a_2}}} = \frac{{{b_1}}}{{{b_2}}} \ne \frac{{{c_1}}}{{{c_2}}}$</span><br />
Therefore, the lines are parallel.</p>
Correct Answer: C