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The problem is asking for the value of \( QR \) given some conditions. Here's how we can solve it:

### Problem Information:
- Points \( P \), \( Q \), \( R \), and \( S \) are collinear.
- Point \( Q \) is between \( P \) and \( R \).
- Point \( R \) is between \( Q \) and \( S \).
- \( PQ = RS \).
- \( PS = 25 \) and \( PR = 19 \).
- We are asked to find \( QR \).

### Breakdown:

- We know the total length \( PS = 25 \), which is the length from \( P \) to \( S \).
- \( PR = 19 \), which is the length from \( P \) to \( R \).
- We are told that \( PQ = RS \). This means that the segment from \( P \) to \( Q \) is equal to the segment from \( R \) to \( S \).

Let's define the unknowns:

- Let \( PQ = x \), which implies \( RS = x \) as well.
- We also know the total length from \( P \) to \( S \) is the sum of the segments \( PQ + QR + RS \). Therefore, \( PS = PQ + QR + RS = x + QR + x = 2x + QR \).

Using the information provided:
1. From \( PS = 25 \), we have the equation \( 2x + QR = 25 \).
2. From \( PR = 19 \), we know that \( PR = PQ + QR = x + QR = 19 \), which gives us the equation \( x + QR = 19 \).

### Solving the System of Equations:

We now have two equations:
1. \( 2x + QR = 25 \)
2. \( x + QR = 19 \)

We can solve these simultaneously.

1. First, solve the second equation for \( x \):
   \[
   x = 19 - QR
   \]

2. Substitute this into the first equation:
   \[
   2(19 - QR) + QR = 25
   \]
   Expanding this:
   \[
   38 - 2QR + QR = 25
   \]
   Simplifying:
   \[
   38 - QR = 25
   \]
   \[
   QR = 38 - 25
   \]
   \[
   QR = 13
   \]

Thus, the value of \( QR \) is \( \boxed{13} \).

Would you like further clarification or details on this solution?

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Here are five related questions to further explore the concepts:

1. What would happen if the value of \( PR \) were different?
2. How would the solution change if \( PQ \neq RS \)?
3. What is the significance of collinearity in this problem?
4. Can you derive the length of \( PQ \) and \( RS \) from the given information?
5. How can we verify that the points are correctly placed according to the given distances?

**Tip**: When solving systems of equations, always start by isolating one variable to simplify substitution into other equations.

Points P, Q, R, and S are collinear. Point Q is between P and R, R is between Q and S, and PQ = RS. If PS = 25 and PR = 19, what is the value of QR?

Learn how to find the value of QR using collinear points and given distances PS = 25, PR = 19, with PQ = RS. This problem involves algebraic manipulation and the segment addition postulate. Suitable for students in Grades 7-9.

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