Question

The graph of the system of equations is given below:

Red Line: y=3+2

Blue Line: y=-x+6

image.png
Part I: Using the graph, determine the solution to the system and explain how you know. Write the solution as an ordered pair
.

Answer 1

To find the solution to the system of equations given by the red line \(y = 3 + 2\) (which simplifies to \(y = 5\)) and the blue line \(y = -x + 6\), we can determine the intersection point of the two lines on the graph.

1. **Identify the equations**:
- The red line has a constant \(y\) value of \(5\) (horizontal line).
- The blue line has a negative slope and \(y\)-intercept of \(6\).

2. **Plotting the lines**:
- The red line will be a horizontal line at \(y = 5\).
- The blue line will decrease as \(x\) increases, starting at \(y = 6\) when \(x = 0\) and crossing the \(y = 5\) line at some \(x\) value.

3. **Finding the intersection**:
- Set \(y = 5\) (from the red line) into the blue line's equation:
\[
5 = -x + 6
\]
Rearranging gives:
\[
-x = 5 - 6 \\
-x = -1 \\
x = 1
\]

4. **Coordinates of the intersection point**:
- The intersection point can then be written as the ordered pair \((1, 5)\).

5. **Conclusion**:
- The solution to the system of equations is the point where the two lines intersect. Since the red line is constant at \(y = 5\) and the blue line is calculated to intersect at \(x = 1\), the solution to the system is:
\[
\boxed{(1, 5)}
\]

This is how we determine the solution: by finding where the lines intersect on the graph and confirming that the coordinates of that intersection satisfy both equations.