Question

Solving the system y=−x+3 and x+y=8 by graphing shows that there is no solution to the system. Is this statement true or false? Explain

Answer 1

To determine whether the statement is true or false, we need to analyze the two equations given:

1. The first equation is \( y = -x + 3 \).
2. The second equation is \( x + y = 8 \).

Let's rewrite the second equation in slope-intercept form (y = mx + b):

1. Start with \( x + y = 8 \).
2. Subtract \( x \) from both sides: \( y = -x + 8 \).

Now, we have the two equations:

1. \( y = -x + 3 \)
2. \( y = -x + 8 \)

Next, we can graph these two lines.

### Line 1: \( y = -x + 3 \)
- This line has a slope of -1 and a y-intercept of 3. It crosses the y-axis at (0, 3).

### Line 2: \( y = -x + 8 \)
- This line also has a slope of -1 and a y-intercept of 8. It crosses the y-axis at (0, 8).

### Analyzing the Graph
Both lines have the same slope (-1), which means they are parallel. However, the y-intercepts are different (3 and 8), indicating that these lines will never intersect.

### Conclusion
Since the two lines are parallel and will never meet, this system of equations has no solution.

Thus, the statement that "solving the system \( y = -x + 3 \) and \( x + y = 8 \) by graphing shows that there is no solution to the system" is **true**.