To solve this problem, we need to find the values of [tex]\( x \)[/tex], [tex]\( EF \)[/tex], and [tex]\( FG \)[/tex] given the equations [tex]\( EF = 2x - 12 \)[/tex], [tex]\( FG = 3x - 15 \)[/tex], and [tex]\( EG = 23 \)[/tex].
Let's break it down step by step:
1. Write down the given equations:
[tex]\[
EF = 2x - 12
\][/tex]
[tex]\[
FG = 3x - 15
\][/tex]
[tex]\[
EG = 23
\][/tex]
2. Express the length [tex]\( EG \)[/tex] in terms of [tex]\( x \)[/tex]:
Since [tex]\( E \)[/tex], [tex]\( F \)[/tex], and [tex]\( G \)[/tex] are collinear points, [tex]\( EG \)[/tex] is the sum of [tex]\( EF \)[/tex] and [tex]\( FG \)[/tex]:
[tex]\[
EG = EF + FG
\][/tex]
Substituting the given expressions for [tex]\( EF \)[/tex] and [tex]\( FG \)[/tex]:
[tex]\[
23 = (2x - 12) + (3x - 15)
\][/tex]
3. Simplify the equation:
Combine like terms on the right-hand side:
[tex]\[
23 = 2x - 12 + 3x - 15
\][/tex]
[tex]\[
23 = 5x - 27
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
Isolate [tex]\( x \)[/tex] by adding 27 to both sides of the equation:
[tex]\[
23 + 27 = 5x
\][/tex]
[tex]\[
50 = 5x
\][/tex]
Divide by 5:
[tex]\[
x = 10
\][/tex]
5. Calculate [tex]\( EF \)[/tex]:
Substitute [tex]\( x = 10 \)[/tex] back into the equation for [tex]\( EF \)[/tex]:
[tex]\[
EF = 2x - 12
\][/tex]
[tex]\[
EF = 2(10) - 12
\][/tex]
[tex]\[
EF = 20 - 12
\][/tex]
[tex]\[
EF = 8
\][/tex]
6. Calculate [tex]\( FG \)[/tex]:
Substitute [tex]\( x = 10 \)[/tex] back into the equation for [tex]\( FG \)[/tex]:
[tex]\[
FG = 3x - 15
\][/tex]
[tex]\[
FG = 3(10) - 15
\][/tex]
[tex]\[
FG = 30 - 15
\][/tex]
[tex]\[
FG = 15
\][/tex]
Therefore, the values are:
- [tex]\( x = 10 \)[/tex]
- [tex]\( EF = 8 \)[/tex]
- [tex]\( FG = 15 \)[/tex]
So, the correct answer is:
[tex]\[
x=10, EF=8, FG=15
\][/tex]
