Question

Find the value of the variable and YZ if y is between X and Z.

XY = 3a-4, YZ = 6a + 2, XZ = 5a + 22

Answer 1

To solve for the values of \( y \), \( YZ \), and the variable \( a \), we can use the information given in the problem. We know that \( Y \) is between \( X \) and \( Z \), and the segments relate to the entire segment as follows:

\[
XZ = XY + YZ
\]

From the problem, we have:

- \( XY = 3a - 4 \)
- \( YZ = 6a + 2 \)
- \( XZ = 5a + 22 \)

Substituting these expressions into the equation \( XZ = XY + YZ \):

\[
5a + 22 = (3a - 4) + (6a + 2)
\]

Now, simplifying the right side:

\[
5a + 22 = 3a - 4 + 6a + 2
\]
\[
5a + 22 = 9a - 2
\]

Now, we rearrange the equation to isolate \( a \):

\[
5a + 22 + 2 = 9a
\]
\[
5a + 24 = 9a
\]
\[
24 = 9a - 5a
\]
\[
24 = 4a
\]
\[
a = \frac{24}{4} = 6
\]

Now that we have the value of \( a \), we can find \( XY \) and \( YZ \):

1. For \( XY \):
\[
XY = 3a - 4 = 3(6) - 4 = 18 - 4 = 14
\]

2. For \( YZ \):
\[
YZ = 6a + 2 = 6(6) + 2 = 36 + 2 = 38
\]

Finally, we can summarize the results:

- Value of \( a \): \( 6 \)
- Value of \( XY \): \( 14 \)
- Value of \( YZ \): \( 38 \)

Thus, the final answers are:

\[
a = 6, \quad YZ = 38
\]