To find the solutions to the system of nonlinear equations given by:
[tex]\[
\begin{cases}
y = -5x - 5 \\
y = x^2 - 5
\end{cases}
\][/tex]
we need to solve these equations simultaneously. Here's a detailed, step-by-step process to find the solutions:
1. Set the two equations equal to each other:
Since both equations are equal to [tex]\( y \)[/tex], we can set the right-hand side of the first equation equal to the right-hand side of the second equation:
[tex]\[
-5x - 5 = x^2 - 5
\][/tex]
2. Simplify the equation:
To solve for [tex]\( x \)[/tex], we want to combine like terms and set the equation to zero:
[tex]\[
x^2 + 5x = 0
\][/tex]
3. Factor the quadratic equation:
Factor out the common term [tex]\( x \)[/tex] from the quadratic expression:
[tex]\[
x(x + 5) = 0
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
We find the solutions for [tex]\( x \)[/tex] by setting each factor equal to zero:
[tex]\[
x = 0 \quad \text{or} \quad x + 5 = 0
\][/tex]
This gives us:
[tex]\[
x = 0 \quad \text{or} \quad x = -5
\][/tex]
5. Find the corresponding [tex]\( y \)[/tex] values:
Substitute the [tex]\( x \)[/tex]-values back into one of the original equations to find the corresponding [tex]\( y \)[/tex]-values. We can use the first equation [tex]\( y = -5x - 5 \)[/tex]:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
y = -5(0) - 5 = -5
\][/tex]
- For [tex]\( x = -5 \)[/tex]:
[tex]\[
y = -5(-5) - 5 = 25 - 5 = 20
\][/tex]
6. List the solutions as ordered pairs:
The solutions to the system of equations are:
[tex]\[
(x, y) = \left\{ (-5, 20), (0, -5) \right\}
\][/tex]
Therefore, the solutions to the given system of equations are:
[tex]\[
(x, y) = \left\{ (-5, 20), (0, -5) \right\}
\][/tex]
Or as a list of ordered pairs:
[tex]\[
[(-5.0, 20.0), (0.0, -5.0)]
\][/tex]
