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The question provided involves solving a system of linear equations by graphing. The two equations given are:

1. \( y = x^2 - 6x + 8 \) (a quadratic equation)
2. \( y = -x + 4 \) (a linear equation)

### To solve graphically:

1. **Graph the quadratic equation \( y = x^2 - 6x + 8 \)**:
   - This is a parabola. To find its vertex and intercepts:
     - The **vertex** is calculated using \( x = -\frac{b}{2a} \), where \( a = 1 \), \( b = -6 \). So, \( x = \frac{6}{2} = 3 \).
     - Substituting \( x = 3 \) into the equation gives \( y = (3)^2 - 6(3) + 8 = 9 - 18 + 8 = -1 \). Thus, the vertex is \( (3, -1) \).
     - Find the **x-intercepts** by solving \( x^2 - 6x + 8 = 0 \), which factors into \( (x - 4)(x - 2) = 0 \). So, the x-intercepts are \( x = 2 \) and \( x = 4 \).
     - The **y-intercept** occurs when \( x = 0 \), giving \( y = 8 \).

2. **Graph the linear equation \( y = -x + 4 \)**:
   - This is a straight line with a slope of -1 and a y-intercept of 4.

### Finding the intersection points:
- The intersection points of the two graphs represent the solutions to the system of equations.
- Setting \( x^2 - 6x + 8 = -x + 4 \), rearrange to:
  \[
  x^2 - 5x + 4 = 0
  \]
  Factor this equation:
  \[
  (x - 4)(x - 1) = 0
  \]
  So, \( x = 4 \) and \( x = 1 \).

- Substituting these values back into the linear equation \( y = -x + 4 \):
  - For \( x = 4 \), \( y = -4 + 4 = 0 \). Thus, one solution is \( (4, 0) \).
  - For \( x = 1 \), \( y = -1 + 4 = 3 \). Thus, the other solution is \( (1, 3) \).

### Solution:
The system has two solutions: \( (4, 0) \) and \( (1, 3) \).

Thus, the correct answer is **C. (1,3) and (4,0)**.

Let me know if you want more details on the solution! Here are 5 related questions to deepen your understanding:

1. How do you determine the vertex of a quadratic equation?
2. How do you find the x- and y-intercepts of a quadratic function?
3. How do you solve a system of equations algebraically?
4. What is the significance of the intersection points of two graphs?
5. How can you verify solutions graphically and algebraically?

**Tip:** Always double-check solutions by substituting them back into the original equations!

Which represents the solution(s) of the system of equations y = x^2 - 6x + 8 and y = -x + 4? Determine the solution set by graphing.

Learn how to solve a system of equations involving a quadratic equation (y = x^2 - 6x + 8) and a linear equation (y = -x + 4) by graphing. Detailed steps include graphing, finding intercepts, and solving by factoring. Suitable for students in Grades 8-10.

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