A line that includes the points (-1,-7) and (8,z) has a slope of 7/9. What is the vaule of z

To find the value of \( z \) for the point \((8, z)\) on the line with a slope of \( \frac{7}{9} \), we can use the slope formula:

\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Given:
- The first point \((x_1, y_1) = (-1, -7)\)
- The second point \((x_2, y_2) = (8, z)\)
- Slope \( m = \frac{7}{9} \)

Substituting these values into the slope formula:

\[
\frac{7}{9} = \frac{z - (-7)}{8 - (-1)}
\]

Simplifying the denominator:

\[
\frac{7}{9} = \frac{z + 7}{8 + 1} = \frac{z + 7}{9}
\]

Now, solve for \( z \) by multiplying both sides by 9:

\[
7 = z + 7
\]

Finally, subtract 7 from both sides:

\[
z = 0
\]

So, the value of \( z \) is \( \boxed{0} \).

Would you like further details or any additional questions?

### Relative Questions:
1. How would the equation change if the slope were different?
2. Can you determine the equation of the line given these two points?
3. What happens to the value of \(z\) if the x-coordinate of the second point changes?
4. How would you find the midpoint between the points \((-1,-7)\) and \((8,z)\)?
5. What would be the slope if \( z \) were 5 instead?

### Tip:
When dealing with linear equations and slope, it's often helpful to sketch a quick graph to visualize the relationship between the points and verify your calculations.

Learn how to calculate the value of z in a line equation passing through points (-1, -7) and (8, z) with a slope of 7/9. Follow a step-by-step solution using the slope formula and algebraic manipulation. Suitable for high school students studying linear equations and slope.

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