X-y=7 xy=-10

To solve the system of equations given by \(x - y = 7\) and \(xy = -10\), we can start by solving the first equation for \(x\): 1. From \(x - y = 7\), we can express \(x\) as: \[ x = y + 7 \] 2. Now, substitute this expression for \(x\) into the second equation \(xy = -10\): \[ (y + 7)y = -10 \] 3. Expanding this gives: \[ y^2 + 7y = -10 \] 4. Rearranging the equation to set it to 0, we have: \[ y^2 + 7y + 10 = 0 \] 5. Now we can factor this quadratic equation: \[ (y + 2)(y + 5) = 0 \] 6. Setting each factor to zero gives the possible values for \(y\): \[ y + 2 = 0 \quad \Rightarrow \quad y = -2 \] \[ y + 5 = 0 \quad \Rightarrow \quad y = -5 \] 7. Now, we can substitute these values back into \(x = y + 7\) to find the corresponding values for \(x\): - If \(y = -2\): \[ x = -2 + 7 = 5 \] - If \(y = -5\): \[ x = -5 + 7 = 2 \] 8. Thus, we have two pairs of solutions: \[ (x, y) = (5, -2) \quad \text{and} \quad (x, y) = (2, -5) \] These are the solutions to the given system of equations.