Question 2

We compare the ratios of coefficients to find the nature of the lines.

Step 1 — Understand the conditions

Let the two linear equations be $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$.

We use these conditions:

If $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$, the lines intersect at one point. If $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$, the lines are parallel. If $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, the lines are coincident.

Step 2 — Solve part (i)

The given equations are $5x - 4y + 8 = 0$ and $7x + 6y - 9 = 0$.

Here, $a_1 = \mathbf{5}$, $b_1 = \mathbf{-4}$, $c_1 = \mathbf{8}$.

And $a_2 = \mathbf{7}$, $b_2 = \mathbf{6}$, $c_2 = \mathbf{-9}$.

Let's find the ratios.

$\frac{a_1}{a_2} = \frac{5}{7}$

$\frac{b_1}{b_2} = \frac{-4}{6}$

$= \frac{-2}{3}$

We compare $\frac{5}{7}$ and $\frac{-2}{3}$.

$\frac{5}{7} \neq \frac{-2}{3}$

So, $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$.

$\boxed{\text{Intersecting lines}}$

Step 3 — Solve part (ii)

The given equations are $9x + 3y + 12 = 0$ and $18x + 6y + 24 = 0$.

Here, $a_1 = \mathbf{9}$, $b_1 = \mathbf{3}$, $c_1 = \mathbf{12}$.

And $a_2 = \mathbf{18}$, $b_2 = \mathbf{6}$, $c_2 = \mathbf{24}$.

Let's find the ratios.

$\frac{a_1}{a_2} = \frac{9}{18}$

$= \frac{1}{2}$

$\frac{b_1}{b_2} = \frac{3}{6}$

$= \frac{1}{2}$

$\frac{c_1}{c_2} = \frac{12}{24}$

$= \frac{1}{2}$

We compare the ratios.

$\frac{1}{2} = \frac{1}{2} = \frac{1}{2}$

So, $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$.

$\boxed{\text{Coincident lines}}$

Step 4 — Solve part (iii)

The given equations are $6x - 3y + 10 = 0$ and $2x - y + 9 = 0$.

Here, $a_1 = \mathbf{6}$, $b_1 = \mathbf{-3}$, $c_1 = \mathbf{10}$.

And $a_2 = \mathbf{2}$, $b_2 = \mathbf{-1}$, $c_2 = \mathbf{9}$.

Let's find the ratios.

$\frac{a_1}{a_2} = \frac{6}{2}$

$= 3$

$\frac{b_1}{b_2} = \frac{-3}{-1}$

$= 3$

$\frac{c_1}{c_2} = \frac{10}{9}$

We compare the ratios.

$3 = 3 \neq \frac{10}{9}$

So, $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$.

$\boxed{\text{Parallel lines}}$

Answer

(i) The lines intersect at a point. (ii) The lines are coincident. (iii) The lines are parallel.