What is the graphical interpretation of the solution to this system of equations?

The graphical interpretation of the solution to a system of linear equations involves plotting the lines represented by each equation on a coordinate plane. The solution is the point where these two lines intersect.

### Step 1: Interpret the equations as lines
You have two equations:

1. \( y = -2x + 9 \)
2. \( y = -\frac{13}{2} \)

- The first equation, \( y = -2x + 9 \), is a line with a slope of \(-2\) and a y-intercept at \( (0, 9) \).
- The second equation, \( y = -\frac{13}{2} \), is a horizontal line, because the y-value is constant. It is located at \( y = -\frac{13}{2} \) (or \( y = -6.5 \)).

### Step 2: Plot the lines
- **Line 1**: Starts at \( (0, 9) \) and has a slope of \(-2\), meaning it goes down 2 units for every 1 unit it moves to the right.
- **Line 2**: Is horizontal and crosses the y-axis at \( y = -6.5 \).

### Step 3: Identify the intersection
- The intersection point of these two lines represents the solution to the system of equations. This point is where both equations are satisfied simultaneously.

From the algebraic solution, we found that the lines intersect at the point \( \left(\frac{31}{4}, -\frac{13}{2}\right) \) or \( \left(7.75, -6.5\right) \).

### Conclusion
Graphically, the solution \( \left(7.75, -6.5\right) \) is the point where the line \( y = -2x + 9 \) and the line \( y = -\frac{13}{2} \) intersect. This point is the only solution to the system, as it satisfies both equations.

Would you like further details or have any questions?

Here are 8 related questions to explore:
1. How do you plot linear equations on a graph?
2. What is the significance of the slope in a linear equation?
3. How do you determine the slope and y-intercept from an equation in slope-intercept form?
4. What happens if two lines in a system of equations are parallel?
5. How do you interpret the intersection of two lines in terms of real-world applications?
6. What does it mean if the two lines in a system coincide (are the same)?
7. How do you graphically determine if a system has no solution?
8. How can graphing calculators be used to solve systems of equations? 

**Tip:** Graphing the equations in a system can help visualize the solution, especially when checking the correctness of your algebraic solution.

Explore the graphical interpretation of the solution to a system of linear equations through detailed examples and explanations. Understand how to plot linear equations on a graph, identify intersection points, and interpret their significance. Suitable for high school students and anyone interested in understanding coordinate geometry.

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