Common Solutions for the System of Linear Equations

Given Equations:
- 5x + 4y + 6 = 0
- 10x + 8y = 12

Explanation:
To find the common solution for the system of linear equations, we need to solve the two equations simultaneously.

Step 1: Simplify the Equations
- 5x + 4y + 6 = 0 can be rewritten as 5x + 4y = -6
- 10x + 8y = 12 can be left as it is

Step 2: Solve the Equations
By solving these two equations simultaneously, we can find the values of x and y that satisfy both equations. If the equations are consistent and independent, they will have a unique solution. If they are inconsistent, they will have no solution. If they are consistent but dependent, they will have infinitely many solutions.

Step 3: Determine the Number of Solutions
- In this case, when we solve the equations, we find that they are dependent, meaning they represent the same line. This implies that they have infinitely many common solutions.
- Therefore, the system of linear equations 5x + 4y + 6 = 0 and 10x + 8y = 12 has infinitely many common solutions.
By following these steps, we can determine the number of common solutions for the given system of linear equations.