https://file.aiquickdraw.com/mathbot/file/1725673242639lykt5rcv.jpg

In this problem, you are asked to find the equation for a line \(\ell_z\) (dashed in red), and the z-coordinates of points \(Q\) and \(P\) are related to that line. Here's how to approach solving it step by step based on the information visible in the image:

1. The parabola \(y = z^2 - 1\) is given (dashed in blue) and the red dashed line \(\ell_z\) intersects the parabola at specific points \(Q\) and \(P\).
2. The goal is to find the equation for the red dashed line \(\ell_z\), given some information about the intersection points with the parabola.

### Approach:
The equation of the red dashed line appears to be in the form of \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Based on the figure and the algebraic setup visible in the image, it seems that the slope \(m = 1\) and the equation of the red dashed line might be \(y = z - 2\).

However, to confirm this and solve the exact values, I'd need more detailed information about the points or the instructions provided above the graph. 

### Next Steps:
- Can you confirm or clarify the problem's wording? It seems to be about finding the equation for the dashed red line based on the relationship between the coordinates of \(P\), \(Q\), and the parabola.

Would you like further details or clarification on specific steps in the solution?

### Related Questions:
1. How do we determine the slope of a line from two points on a graph?
2. What is the general form of a line equation, and how do you use it with given points?
3. How do you solve for the intersection points between a parabola and a line?
4. What role do the z-coordinates of points \(P\) and \(Q\) play in this equation?
5. Can you explore how the properties of the parabola affect the solution for the line?

**Tip:** When working with line equations, knowing the slope and one point on the line can allow you to quickly write the equation in slope-intercept form.

Find the equation for the line ℓ_z (dashed in red) in the figure below. Note the z-coordinates of the point Q, z_Q, are coordinated at the point P as z_P, where the parabola (dashed in blue) has equation y = z^2 - 1.

This problem involves finding the equation of a line that intersects a given parabola y = z^2 - 1 at specific points. It covers the equation of a line in slope-intercept form and finding intersections between a line and a parabola. Suitable for high school students in Grades 10-12.

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