Sure! Let's determine the values of [tex]\( x \)[/tex] that satisfy the given system of equations:
[tex]\[
\begin{cases}
y = x^2 + 8x - 5 \\
y = 8x - 4
\end{cases}
\][/tex]
### Step 1: Set the equations equal to each other
Since both expressions equal [tex]\( y \)[/tex], we can set them equal to each other:
[tex]\[
x^2 + 8x - 5 = 8x - 4
\][/tex]
### Step 2: Simplify the equation
Subtract [tex]\( 8x \)[/tex] from both sides of the equation to eliminate [tex]\( 8x \)[/tex]:
[tex]\[
x^2 + 8x - 8x - 5 = 8x - 8x - 4
\][/tex]
This simplifies to:
[tex]\[
x^2 - 5 = -4
\][/tex]
Next, add 4 to both sides to isolate the [tex]\( x^2 \)[/tex] term:
[tex]\[
x^2 - 5 + 4 = -4 + 4
\][/tex]
This simplifies further to:
[tex]\[
x^2 - 1 = 0
\][/tex]
### Step 3: Solve the quadratic equation
Add 1 to both sides:
[tex]\[
x^2 = 1
\][/tex]
Now, take the square root of both sides:
[tex]\[
x = \pm 1
\][/tex]
So, the solutions for [tex]\( x \)[/tex] are [tex]\( x = 1 \)[/tex] and [tex]\( x = -1 \)[/tex].
### Step 4: Determine corresponding [tex]\( y \)[/tex] values
We need to find the corresponding [tex]\( y \)[/tex] values for [tex]\( x = 1 \)[/tex] and [tex]\( x = -1 \)[/tex] using one of the original equations, e.g., [tex]\( y = 8x - 4 \)[/tex].
- For [tex]\( x = 1 \)[/tex]:
[tex]\[
y = 8(1) - 4 = 8 - 4 = 4
\][/tex]
Thus, one solution is [tex]\( (1, 4) \)[/tex].
- For [tex]\( x = -1 \)[/tex]:
[tex]\[
y = 8(-1) - 4 = -8 - 4 = -12
\][/tex]
Thus, another solution is [tex]\( (-1, -12) \)[/tex].
### Conclusion
The values of [tex]\( x \)[/tex] that satisfy the system of equations are [tex]\( x = 1 \)[/tex] and [tex]\( x = -1 \)[/tex], with respective [tex]\( y \)[/tex] values of 4 and -12. Therefore, the solutions to the system of equations are:
[tex]\[ (-1, -12) \text{ and } (1, 4) \][/tex]
