Give the answer of m= and b= and also the system of y= -x + 6 and y= -x + 1

To find the values of \( m \) (the slope) and \( b \) (the y-intercept) for both lines in the system of equations:

The given equations are:

1. \( y = -x + 6 \)
2. \( y = -x + 1 \)

### For the first equation \( y = -x + 6 \):

- The slope \( m \) is the coefficient of \( x \), which is \( -1 \).
- The y-intercept \( b \) is the constant term, which is \( 6 \).

So, for \( y = -x + 6 \):
- \( m = -1 \)
- \( b = 6 \)

### For the second equation \( y = -x + 1 \):

- The slope \( m \) is again \( -1 \) (since the coefficient of \( x \) is still \( -1 \)).
- The y-intercept \( b \) is \( 1 \).

So, for \( y = -x + 1 \):
- \( m = -1 \)
- \( b = 1 \)

### System of Equations:
You can now express the system as:
\[
\begin{cases}
y = -x + 6 \\
y = -x + 1
\end{cases}
\]

These two lines are parallel because they have the same slope \( m = -1 \) but different y-intercepts \( b = 6 \) and \( b = 1 \). Therefore, they will never intersect, meaning there is no solution to this system.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How do you determine the point of intersection of two lines if they intersect?
2. What does it mean geometrically if two lines have the same slope but different y-intercepts?
3. How can you write the equation of a line if you are given two points on it?
4. What changes to the equation would make these two lines intersect?
5. Can two lines with different slopes ever be parallel?

**Tip:** When two lines have the same slope but different y-intercepts, they are parallel and never intersect.

Learn how to find the slope (m) and y-intercept (b) for the linear equations y= -x + 6 and y= -x + 1. Understand why these lines are parallel and explore key algebra concepts. Suitable for students in Grades 8-10.

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