Questions: H -- G -- F -- E 14 2x+5 G 3x-2

Transcript text: 10 H -- G -- F -- E 14 2x+5 G 3x-2

Solution

Solution Steps

Step 1: Understand the Problem

We are given a line segment \(HE\) with points \(G\) and \(F\) on it. The distances are labeled as follows:

\(HG = 10\)
\(GF = 3x - 2\)
\(HF = 14\)
\(FE = 2x + 5\)

Step 2: Set Up the Equation

Since \(H, G, F, E\) are collinear, the sum of the segments \(HG\), \(GF\), and \(FE\) should equal the total length \(HE\). We can write the equation as: \[ HG + GF + FE = HE \]

Step 3: Substitute Known Values

Substitute the given values into the equation: \[ 10 + (3x - 2) + (2x + 5) = 14 + (2x + 5) \]

Step 4: Simplify the Equation

Combine like terms on the left side: \[ 10 + 3x - 2 + 2x + 5 = 14 + 2x + 5 \] \[ 10 - 2 + 5 + 3x + 2x = 14 + 2x + 5 \] \[ 13 + 5x = 19 + 2x \]

Step 5: Solve for \(x\)

Isolate \(x\) by moving all \(x\)-terms to one side and constants to the other: \[ 5x - 2x = 19 - 13 \] \[ 3x = 6 \] \[ x = 2 \]

Final Answer

The value of \(x\) is \(2\).

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