To solve for [tex]\( YZ \)[/tex] given the information:
1. Identify the given expressions:
- [tex]\( XY = 4x + 4 \)[/tex]
- [tex]\( YZ = 5x \)[/tex]
- [tex]\( XZ = 58 \)[/tex]
2. Use the fact that the total distance [tex]\( XZ \)[/tex] is the sum of [tex]\( XY \)[/tex] and [tex]\( YZ \)[/tex]:
[tex]\[
XZ = XY + YZ
\][/tex]
3. Substitute the expressions for [tex]\( XY \)[/tex] and [tex]\( YZ \)[/tex] into the equation [tex]\( XZ = 58 \)[/tex]:
[tex]\[
58 = (4x + 4) + 5x
\][/tex]
4. Combine like terms:
[tex]\[
58 = 4x + 4 + 5x \quad \Rightarrow \quad 58 = 9x + 4
\][/tex]
5. Isolate [tex]\( x \)[/tex] by first subtracting 4 from both sides of the equation:
[tex]\[
58 - 4 = 9x \quad \Rightarrow \quad 54 = 9x
\][/tex]
6. Solve for [tex]\( x \)[/tex] by dividing both sides by 9:
[tex]\[
x = \frac{54}{9} \quad \Rightarrow \quad x = 6
\][/tex]
7. Substitute [tex]\( x = 6 \)[/tex] back into the expression for [tex]\( YZ \)[/tex]:
[tex]\[
YZ = 5x \quad \Rightarrow \quad YZ = 5 \cdot 6 \quad \Rightarrow \quad YZ = 30
\][/tex]
So, the value of [tex]\( YZ \)[/tex] is:
[tex]\[
YZ = 30
\][/tex]
