To find the values of [tex]\( n \)[/tex] and [tex]\( YZ \)[/tex] given the equations [tex]\( XY = 4n + 3 \)[/tex], [tex]\( YZ = 2n - 7 \)[/tex], and [tex]\( XZ = 20 \)[/tex], we start by using the relationship that when [tex]\( Y \)[/tex] is between [tex]\( X \)[/tex] and [tex]\( Z \)[/tex], the sum of the distances [tex]\( XY \)[/tex] and [tex]\( YZ \)[/tex] should equal [tex]\( XZ \)[/tex].
### Step-by-Step Solution:
1. Write down the given equations:
[tex]\[
XY = 4n + 3
\][/tex]
[tex]\[
YZ = 2n - 7
\][/tex]
[tex]\[
XZ = 20
\][/tex]
2. Formulate the equation from the total distance:
Since [tex]\( XY + YZ = XZ \)[/tex], we write:
[tex]\[
(4n + 3) + (2n - 7) = 20
\][/tex]
3. Combine like terms:
Simplify the left side of the equation:
[tex]\[
4n + 3 + 2n - 7 = 20
\][/tex]
[tex]\[
6n - 4 = 20
\][/tex]
4. Solve for [tex]\( n \)[/tex]:
Add 4 to both sides to isolate the term with [tex]\( n \)[/tex]:
[tex]\[
6n - 4 + 4 = 20 + 4
\][/tex]
[tex]\[
6n = 24
\][/tex]
Divide both sides by 6:
[tex]\[
n = 4
\][/tex]
5. Calculate [tex]\( YZ \)[/tex]:
Substitute [tex]\( n = 4 \)[/tex] back into the equation for [tex]\( YZ \)[/tex]:
[tex]\[
YZ = 2n - 7
\][/tex]
[tex]\[
YZ = 2(4) - 7
\][/tex]
[tex]\[
YZ = 8 - 7
\][/tex]
[tex]\[
YZ = 1
\][/tex]
### Final Answers:
[tex]\[
n = 4
\][/tex]
[tex]\[
YZ = 1
\][/tex]
