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We are given that \( Y \) is between \( X \) and \( Z \), and the following information:

\[
XY = 3x + 1, \quad YZ = 2x - 2, \quad XZ = 84
\]

The problem requires us to find:

1. \( x \)
2. \( XY \)
3. \( YZ \)

### Step 1: Solve for \( x \)

Since \( Y \) is between \( X \) and \( Z \), we know that:

\[
XY + YZ = XZ
\]

Substitute the given expressions for \( XY \) and \( YZ \):

\[
(3x + 1) + (2x - 2) = 84
\]

Simplify the equation:

\[
5x - 1 = 84
\]

Add 1 to both sides:

\[
5x = 85
\]

Divide by 5:

\[
x = 17
\]

### Step 2: Calculate \( XY \)

Now substitute \( x = 17 \) into the expression for \( XY \):

\[
XY = 3(17) + 1 = 51 + 1 = 52
\]

### Step 3: Calculate \( YZ \)

Substitute \( x = 17 \) into the expression for \( YZ \):

\[
YZ = 2(17) - 2 = 34 - 2 = 32
\]

### Final Answers:
1. \( x = 17 \)
2. \( XY = 52 \)
3. \( YZ = 32 \)

Would you like more details on any of the steps? Here are 5 related questions to deepen understanding:

1. What if \( XY + YZ \) did not equal \( XZ \)? How would that change the problem?
2. How can we verify the solution for \( x \) is correct?
3. What is the significance of placing \( Y \) between \( X \) and \( Z \)?
4. How would the problem change if \( YZ = 2x + 2 \) instead?
5. Can this type of equation be solved graphically? How?

**Tip:** When solving equations like this, always ensure you simplify and combine like terms carefully before solving for the unknown!

Y is between X and Z, XY = 3x + 1, YZ = 2x - 2, XZ = 84. Find each of the following: 1. x = ?, 2. XY = ?, 3. YZ = ?

Learn how to solve a linear equation involving XY, YZ, and XZ where Y is between X and Z. This math problem covers algebraic manipulation and geometric reasoning. Suitable for students in Grades 6-8.

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