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.