To find the value of the variable [tex]\( t \)[/tex] and [tex]\( YZ \)[/tex] given the equations [tex]\( XY = 2t + 2 \)[/tex], [tex]\( YZ = 5t - 3 \)[/tex], and [tex]\( XZ = 34 \)[/tex] with [tex]\( Y \)[/tex] being between [tex]\( X \)[/tex] and [tex]\( Z \)[/tex], follow these detailed steps:
1. First, note that since [tex]\( Y \)[/tex] is between [tex]\( X \)[/tex] and [tex]\( Z \)[/tex], we have:
[tex]\[
XY + YZ = XZ
\][/tex]
2. Substitute the given equations for [tex]\( XY \)[/tex], [tex]\( YZ \)[/tex], and [tex]\( XZ \)[/tex] into the equation above:
[tex]\[
(2t + 2) + (5t - 3) = 34
\][/tex]
3. Combine like terms inside the equation:
[tex]\[
2t + 5t + 2 - 3 = 34
\][/tex]
[tex]\[
7t - 1 = 34
\][/tex]
4. Add 1 to both sides of the equation to isolate the term with [tex]\( t \)[/tex] on one side:
[tex]\[
7t = 34 + 1
\][/tex]
[tex]\[
7t = 35
\][/tex]
5. Divide both sides by 7 to solve for [tex]\( t \)[/tex]:
[tex]\[
t = \frac{35}{7}
\][/tex]
[tex]\[
t = 5
\][/tex]
6. Now, substitute the value of [tex]\( t \)[/tex] back into the equation for [tex]\( YZ \)[/tex] to find [tex]\( YZ \)[/tex]:
[tex]\[
YZ = 5t - 3
\][/tex]
[tex]\[
YZ = 5(5) - 3
\][/tex]
[tex]\[
YZ = 25 - 3
\][/tex]
[tex]\[
YZ = 22
\][/tex]
Therefore, the value of [tex]\( t \)[/tex] is [tex]\( 5 \)[/tex] and the value of [tex]\( YZ \)[/tex] is [tex]\( 22 \)[/tex].
[tex]\[
t = 5
\][/tex]
[tex]\[
YZ = 22
\][/tex]
