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, let's analyze the two equations given in the system:

1. The first equation is \( y = -x + 3 \).
2. The second equation is \( x + y = 8 \), which can be rewritten in slope-intercept form (i.e., \( y = mx + b \)) as:
\[ y = -x + 8 \].

Now we have two equations:

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

Next, let's graph both equations:

- For the first equation \( y = -x + 3 \), the y-intercept is at (0, 3) and the slope is -1. This means that the line descends to the right.
- For the second equation \( y = -x + 8 \), the y-intercept is at (0, 8) and the slope is also -1.

Since both lines have the same slope (-1), they are parallel. However, they have different y-intercepts (3 and 8), which means they will never intersect.

Since the lines do not intersect, there is no solution to the system of equations.

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