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To solve this system of equations by finding the points where the graphs intersect, let's analyze each equation:

1. **Equation 1**: \( y = -x + 4 \)
   - This is a linear equation with a slope of \(-1\) and a y-intercept at \( (0, 4) \). 
   - The line will decrease as \( x \) increases.

2. **Equation 2**: \( y = x^2 - 4 \)
   - This is a quadratic equation, representing a parabola that opens upwards with a vertex at \( (0, -4) \).

To find the intersection points, we set the equations equal to each other:

\[
-x + 4 = x^2 - 4
\]

Rearrange all terms to one side:

\[
x^2 + x - 8 = 0
\]

Now, we can solve this quadratic equation using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \( a = 1 \), \( b = 1 \), and \( c = -8 \):

\[
x = \frac{-1 \pm \sqrt{(1)^2 - 4 \cdot 1 \cdot (-8)}}{2 \cdot 1}
\]
\[
x = \frac{-1 \pm \sqrt{1 + 32}}{2}
\]
\[
x = \frac{-1 \pm \sqrt{33}}{2}
\]

Calculating the two values for \( x \):

1. \( x = \frac{-1 + \sqrt{33}}{2} \approx 2.37 \)
2. \( x = \frac{-1 - \sqrt{33}}{2} \approx -3.37 \)

Now, substitute these values of \( x \) back into one of the original equations (e.g., \( y = -x + 4 \)) to find the corresponding \( y \)-values.

For \( x \approx 2.37 \):
\[
y = -2.37 + 4 \approx 1.63
\]

For \( x \approx -3.37 \):
\[
y = -(-3.37) + 4 \approx 7.37
\]

### Solution
The approximate points of intersection are:
1. \( (2.37, 1.63) \)
2. \( (-3.37, 7.37) \)

Would you like further clarification or more details on this solution?

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Here are some related questions for further practice:
1. How would you solve a system involving a linear and cubic equation?
2. How do you interpret the intersection points in terms of real-world applications?
3. What other methods can be used to solve systems of nonlinear equations besides graphing?
4. How do you solve a system of equations if one equation is a circle and the other is a line?
5. What effect does changing the slope in a linear equation have on the intersection points with a quadratic?

**Tip:** When graphing, always identify the key points (like intercepts and vertex) to help guide your plotting.

Solve the linear-exponential system of equations by graphing on your own paper. Express your values as decimals rounded to the nearest hundredth, if necessary.

 y = -x + 4
 y = x^2 - 4

Learn how to solve the system of equations y = -x + 4 and y = x^2 - 4 by graphing. Find the points of intersection to express values as decimals rounded to the nearest hundredth. Suitable for students in Grades 9-11.

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