### Theory:

Find the given system of equations have a consistent or inconsistent solution.

1. $$3x - 5y = -2$$ and $$6x + 7y = 6$$

Solution:

The given equations can be written as $$3x - 5y + 2 = 0$$ and $$6x + 7y - 6 = 0$$.

Here, $$a_1 = 3$$, $$b_1 = -5$$, $$c_1 = 2$$, $$a_2 = 6$$, $$b_2 = 7$$, $$c_2 = -6$$.

Comparing the ratios with $$\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$$, we have:

$$\frac{3}{6} \neq \frac{-5}{7}$$

Simplifying, we get:

$$\frac{1}{2} \neq \frac{-5}{7}$$

Since, $$\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$$, then, the system of equations has a unique solution and is said to be a consistent system.

2. $$6x + 4y = -8$$ and $$3x + 2y = -4$$

Solution:

The given equations can be written as $$6x + 4y + 8 = 0$$ and $$3x + 2y + 4 =0$$.

Here, $$a_1 = 6$$, $$b_1 = 4$$, $$c_1 = 8$$, $$a_2 = 3$$, $$b_2 = 2$$, $$c_2 = 4$$.

Comparing the ratios with $$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$$, we have:

$$\frac{6}{3} = \frac{4}{2} = \frac{8}{4}$$

Simplifying, we get:

$$2 = 2 = 2$$

Since, $$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$$, then, the system of equations has infinitely many solutions and is said to be a consistent system.

3. $$5x + 2y = 7$$ and $$10x + 4y = 6$$

Solution:

The given equations can be written as $$5x + 2y - 7 = 0$$ and $$10x + 4y - 6 = 0$$.

Here, $$a_1 = 5$$, $$b_1 = 2$$, $$c_1 = -7$$, $$a_2 = 10$$, $$b_2 = 4$$, $$c_2 = -6$$.

Comparing the ratios with $$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$$, we have:

$$\frac{5}{10} = \frac{2}{4} \neq \frac{-7}{-6}$$

Simplifying, we get:

$$\frac{1}{2} = \frac{1}{2} \neq \frac{7}{6}$$

Since, $$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$$, then, the system of equations has no solution and is said to be an inconsistent system.