Questions: Determine the number of solutions for the following system of linear equations. If there is only one solution, find the solution. -3x - 5y + 7 = -5 6x + 4y + 1 = 3 -3x - 5y = -12 6x + 4y = 2 Answer: Infinitely Many Solutions Only One Solution No Solution x = y = z =

Solution Steps

To determine the number of solutions for the given system of linear equations, we can use the method of elimination or substitution to solve for the variables \(x\) and \(y\). If the system has a unique solution, we will find specific values for \(x\) and \(y\). If the equations are dependent, there will be infinitely many solutions, and if they are inconsistent, there will be no solution.

1. Simplify the given equations to standard form.
2. Use the elimination method to eliminate one of the variables.
3. Solve for the remaining variable.
4. Substitute back to find the other variable.
5. Check for consistency to determine the number of solutions.

The given system of equations is: \[ \begin{cases} -3x - 5y = -12 \\ 6x + 4y = 2 \end{cases} \]

Using the elimination or substitution method, we solve for \(x\) and \(y\). The solution to the system is: \[ x = -\frac{19}{9}, \quad y = \frac{11}{3} \]

Since we have found specific values for both \(x\) and \(y\), the system has a unique solution.

Final Answer