1. Graphically, the pair of equations
6x – 3y + 10 = 0
2x – y + 9 = 0
represents two lines which are
intersecting at exactly one point.
intersecting at exactly two points.
coincident.
parallel.
Step 1: Write equations in standard form.
Equation 1: \(6x - 3y + 10 = 0\) ⇒ coefficients \(a_1 = 6\), \(b_1 = -3\), \(c_1 = 10\).
Equation 2: \(2x - y + 9 = 0\) ⇒ coefficients \(a_2 = 2\), \(b_2 = -1\), \(c_2 = 9\).
Step 2: Compare the ratios of coefficients.
\(\dfrac{a_1}{a_2} = \dfrac{6}{2} = 3\)
\(\dfrac{b_1}{b_2} = \dfrac{-3}{-1} = 3\)
\(\dfrac{c_1}{c_2} = \dfrac{10}{9}\)
Step 3: Apply the condition for parallel lines.
If \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}\), then the lines are parallel and distinct.
Here, \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = 3\), but \(\dfrac{c_1}{c_2} = \dfrac{10}{9}\), which is not equal to 3.
Conclusion: The two lines are parallel distinct lines.