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The graph provided shows a linear relationship between the distance \( d \) (in kilometers) and the fuel remaining \( f \) (in liters) in Karl's truck. The problem asks how many kilometers Karl had driven when 200 liters of fuel remained in the tank.

From the graph:

- At \( d = 0 \) kilometers, the fuel \( f = 500 \) liters.
- At \( d = 900 \) kilometers, the fuel \( f = 0 \) liters.

This establishes that the relationship between distance and fuel is linear. We can use the slope-intercept form of a line to find the equation of the line.

### Step 1: Determine the slope of the line

The slope \( m \) of a line is given by:

\[
m = \frac{\text{change in } f}{\text{change in } d} = \frac{f_2 - f_1}{d_2 - d_1}
\]

From the graph:
- \( f_1 = 500 \) liters when \( d_1 = 0 \) kilometers.
- \( f_2 = 0 \) liters when \( d_2 = 900 \) kilometers.

Thus, the slope is:

\[
m = \frac{0 - 500}{900 - 0} = \frac{-500}{900} = -\frac{5}{9}
\]

### Step 2: Equation of the line

Using the slope-intercept form \( f = m \cdot d + b \), where \( b \) is the initial amount of fuel (500 liters), the equation becomes:

\[
f = -\frac{5}{9} \cdot d + 500
\]

### Step 3: Solve for when \( f = 200 \) liters

To find the distance \( d \) when \( f = 200 \):

\[
200 = -\frac{5}{9} \cdot d + 500
\]

Solve for \( d \):

\[
200 - 500 = -\frac{5}{9} \cdot d
\]
\[
-300 = -\frac{5}{9} \cdot d
\]
\[
d = \frac{300 \times 9}{5} = 540 \text{ kilometers}
\]

### Conclusion:
Karl had driven 540 kilometers when the fuel tank had exactly 200 liters remaining.

Would you like further explanation or have any questions? Here are some additional questions to explore:

1. What is the interpretation of the slope in this context?
2. How would the equation change if Karl started with 600 liters instead of 500?
3. What would the fuel remaining be after driving 300 kilometers?
4. How far can Karl drive before running out of fuel if he only had 300 liters to start with?
5. How does the slope of the line relate to fuel efficiency?

**Tip**: When dealing with linear relationships, finding the slope helps you understand how one variable changes in relation to another.

After how many kilometers of driving did the tank have exactly 200 liters of fuel remaining?

Find out how many kilometers Karl drove before his fuel tank had 200 liters left using a linear equation. Learn about slope, intercepts, and solving real-world algebraic problems. Suitable for Grades 9-11.

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