Questions: Find the ordered pair solutions for the system of equations. [ left y=x^2-4 x-5 y=-x-7 right. ]

Solution Steps

To find the ordered pair solutions for the given system of equations, we need to solve the equations simultaneously. We can do this by substituting the expression for \( y \) from the second equation into the first equation and solving for \( x \). Once we have the values of \( x \), we can substitute them back into either equation to find the corresponding \( y \) values.

We are given the system of equations: \[ \begin{cases} y = x^2 - 4x - 5 \\ y = -x - 7 \end{cases} \]

Substitute \( y = -x - 7 \) into \( y = x^2 - 4x - 5 \): \[ -x - 7 = x^2 - 4x - 5 \]

Rearrange the equation to form a standard quadratic equation: \[ x^2 - 4x - 5 + x + 7 = 0 \] \[ x^2 - 3x + 2 = 0 \]

Solve the quadratic equation \( x^2 - 3x + 2 = 0 \): \[ (x - 1)(x - 2) = 0 \] Thus, the solutions for \( x \) are: \[ x = 1 \quad \text{and} \quad x = 2 \]

Substitute \( x = 1 \) and \( x = 2 \) back into \( y = -x - 7 \) to find the corresponding \( y \) values: \[ \text{For } x = 1: \quad y = -1 - 7 = -8 \] \[ \text{For } x = 2: \quad y = -2 - 7 = -9 \]

Final Answer