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The problem shows three collinear points, \( F \), \( G \), and \( H \), on a line with the distances between them expressed algebraically. We are given that:

- \( FH = 16 \)
- The distance \( FG \) is \( 3x - 1 \)
- The distance \( GH \) is \( 2x + 2 \)

Since the points are collinear, the sum of the distances \( FG \) and \( GH \) equals the total distance \( FH \). This gives us the equation:

\[
FG + GH = FH
\]

Substituting the values:

\[
(3x - 1) + (2x + 2) = 16
\]

Now, let me solve this equation for \( x \).

\[
3x - 1 + 2x + 2 = 16
\]
\[
5x + 1 = 16
\]
\[
5x = 16 - 1
\]
\[
5x = 15
\]
\[
x = 3
\]

Now that we have \( x = 3 \), we can find the distances \( FG \) and \( GH \):

\[
FG = 3(3) - 1 = 9 - 1 = 8
\]
\[
GH = 2(3) + 2 = 6 + 2 =

Points F, G, and H are collinear. If FH = 16, FG = 3x - 1, and GH = 2x + 2, find the value of x and the lengths of FG and GH.

Learn how to solve a collinear points problem where the distances FG and GH are given in terms of x, and the total distance FH is provided. This step-by-step solution involves solving a linear equation to find the unknown value of x. Suitable for students in Grades 8-10.

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