Sure, let's solve the given system of equations step-by-step.
The system of equations is:
[tex]\[
\left\{\begin{array}{l}
y = -x - 10 \\
y = x^2 - 5x - 10
\end{array}\right.
\][/tex]
1. Step 1: Set the equations equal to each other
Since both equations are equal to [tex]\( y \)[/tex], we can set the right-hand sides of the equations equal to each other:
[tex]\[
-x - 10 = x^2 - 5x - 10
\][/tex]
2. Step 2: Move all terms to one side of the equation to set it to zero
Add [tex]\( x + 10 \)[/tex] to both sides:
[tex]\[
0 = x^2 - 5x - 10 + x + 10
\][/tex]
3. Step 3: Combine like terms
Simplify the equation:
[tex]\[
0 = x^2 - 4x
\][/tex]
4. Step 4: Factor the quadratic equation
Factor out [tex]\( x \)[/tex]:
[tex]\[
0 = x(x - 4)
\][/tex]
5. Step 5: Solve for [tex]\( x \)[/tex]
Set each factor equal to zero and solve:
[tex]\[
x = 0 \quad \text{or} \quad x = 4
\][/tex]
6. Step 6: Substitute each [tex]\( x \)[/tex] value back into one of the original equations to find [tex]\( y \)[/tex]
Let's use the equation [tex]\( y = -x - 10 \)[/tex]:
For [tex]\( x = 0 \)[/tex]:
[tex]\[
y = -0 - 10 = -10
\][/tex]
So, one solution is [tex]\((0, -10)\)[/tex].
For [tex]\( x = 4 \)[/tex]:
[tex]\[
y = -4 - 10 = -14
\][/tex]
So, another solution is [tex]\((4, -14)\)[/tex].
7. Conclusion:
The solutions to the system of equations are:
[tex]\[
(0, -10) \quad \text{and} \quad (4, -14)
\][/tex]
