Questions: Find the value of W and YZ if Y is between X and Z. X Y=4 w, YZ=6 w, XZ=12 w-8 w= YZ=

Find the value of W and YZ if Y is between X and Z. X Y=4 w, YZ=6 w, XZ=12 w-8 w= YZ=
Transcript text: Find the value of $W$ and $Y Z$ if $Y$ is between $X$ and $Z$. \[ \begin{array}{l} X \mathrm{Y}=4 \mathrm{w}, \mathrm{YZ}=6 \mathrm{w}, \mathrm{XZ}=12 \mathrm{w}-8 \\ \mathrm{w}=\square \\ \mathrm{YZ}=\square \end{array} \]
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Solution

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Solution Steps

To find the value of \( w \) and \( YZ \), we can use the given equations and the fact that \( Y \) is between \( X \) and \( Z \). This implies that the sum of \( XY \) and \( YZ \) should equal \( XZ \).

  1. Set up the equation \( XY + YZ = XZ \).
  2. Substitute the given expressions: \( 4w + 6w = 12w - 8 \).
  3. Solve for \( w \).
  4. Once \( w \) is found, substitute it back into the expression for \( YZ \) to find its value.
Step 1: Set Up the Equation

Given the conditions: \[ XY = 4w, \quad YZ = 6w, \quad XZ = 12w - 8 \] Since \( Y \) is between \( X \) and \( Z \), we have: \[ XY + YZ = XZ \]

Step 2: Substitute and Simplify

Substitute the given expressions into the equation: \[ 4w + 6w = 12w - 8 \] Simplify the equation: \[ 10w = 12w - 8 \]

Step 3: Solve for \( w \)

Rearrange the equation to solve for \( w \): \[ 10w - 12w = -8 \implies -2w = -8 \implies w = 4 \]

Step 4: Calculate \( YZ \)

Substitute \( w = 4 \) into the expression for \( YZ \): \[ YZ = 6w = 6 \times 4 = 24 \]

Final Answer

The value of \( w \) is: \[ \boxed{w = 4} \] The value of \( YZ \) is: \[ \boxed{YZ = 24} \]

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