https://file.aiquickdraw.com/mathbot/file/1726099004229sx6xkv1g.jpg

The image provided contains a problem setup with the following information:

- \( DE = 4x - 16 \)
- \( EF = 8 \)
- \( DF = \frac{7x + 19}{2} \)

The geometry setup is given by the equation:

\[
DE + EF = DF
\]

To solve for \( x \), we can substitute the given expressions into the equation.

### Step 1: Substitute the values into the equation

\[
(4x - 16) + 8 = \frac{7x + 19}{2}
\]

### Step 2: Simplify the left-hand side

\[
4x - 16 + 8 = \frac{7x + 19}{2}
\]

\[
4x - 8 = \frac{7x + 19}{2}
\]

### Step 3: Eliminate the fraction by multiplying both sides by 2

\[
2(4x - 8) = 7x + 19
\]

\[
8x - 16 = 7x + 19
\]

### Step 4: Solve for \( x \)

Subtract \( 7x \) from both sides:

\[
x - 16 = 19
\]

Add 16 to both sides:

\[
x = 35
\]

### Step 5: Find \( DE \) and \( DF \)

- For \( DE \):
\[
DE = 4x - 16 = 4(35) - 16 = 140 - 16 = 124
\]

- For \( DF \):
\[
DF = \frac{7x + 19}{2} = \frac{7(35) + 19}{2} = \frac{245 + 19}{2} = \frac{264}{2} = 132
\]

So, \( x = 35 \), \( DE = 124 \), and \( DF = 132 \), which matches the values in the image.

---

Do you have any further questions or want additional details?

### Related Questions:
1. How would the equation change if \( DF \) were different?
2. How do you derive the equation for segments in a line geometry problem?
3. What happens if \( EF \) were unknown? How would you solve for it?
4. Can you solve for \( x \) using another method like graphing?
5. How does this method generalize to more complex geometric setups?

**Tip:** Always check your algebraic simplifications to avoid errors when working with expressions like fractions.

If DE = 4x - 16, EF = 8, and DF = (7x + 19)/2, solve for x, DE, and DF.

Learn how to solve a geometry problem involving the equation DE + EF = DF with detailed steps. The solution involves algebraic manipulation, linear equations, and the Segment Addition Postulate. This is suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation