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Find the value of [tex]\( n \)[/tex] and [tex]\( YZ \)[/tex] if [tex]\( Y \)[/tex] is between [tex]\( X \)[/tex] and [tex]\( Z \)[/tex].

[tex]\[
\begin{array}{l}
XY = 4n + 3, \quad YZ = 2n - 7, \quad XZ = 20 \\
n = \\
YZ =
\end{array}
\][/tex]

Sagot :

GinnyAnsweridn
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]
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