Questions: Solve the system of equations -7x + 5y = 25, 4x + 7y = -34. How many solutions are there? Select an answer. If there are no solutions, enter DNE below. If there is one solution, enter it below as an ordered pair. If there are infinitely many solutions, enter any ordered pair which is a solution.

Solution Steps

To solve the system of equations, we can use a method such as substitution or elimination. Here, we'll use the elimination method to eliminate one of the variables by adding or subtracting the equations. Once one variable is eliminated, we can solve for the other variable and then back-substitute to find the first variable. Finally, we'll check if the solution satisfies both equations to ensure it's correct.

We start with the system of equations: \[ \begin{align_} -7x + 5y &= 25 \quad \text{(1)} \\ 4x + 7y &= -34 \quad \text{(2)} \end{align_} \]

Using the elimination method, we find the values of \(x\) and \(y\). After solving the system, we find: \[ x = -5, \quad y = -2 \]

We substitute \(x = -5\) and \(y = -2\) back into both equations to verify:

1. For equation (1): \[ -7(-5) + 5(-2) = 35 - 10 = 25 \quad \text{(True)} \]
2. For equation (2): \[ 4(-5) + 7(-2) = -20 - 14 = -34 \quad \text{(True)} \]

Both equations are satisfied, confirming that the solution is correct.

Final Answer