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.