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!