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The problem asks for an equation representing the relationship between the number of miles traveled, \( x \), and the gallons of gas used, \( y \), based on the provided table.

The table is:

\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & 35 & 70 & 105 & 140 & 175 \\
\hline
y & 1 & 2 & 3 & 4 & 5 \\
\hline
\end{array}
\]

To find the equation, notice that this represents a linear relationship. The relationship between \( x \) and \( y \) can be expressed as:

\[
y = mx + b
\]

Where \( m \) is the slope and \( b \) is the y-intercept.

### Step 1: Calculate the slope \( m \)

The slope \( m \) is given by the formula:

\[
m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the first two points, \( (x_1, y_1) = (35, 1) \) and \( (x_2, y_2) = (70, 2) \):

\[
m = \frac{2 - 1}{70 - 35} = \frac{1}{35}
\]

### Step 2: Find the y-intercept \( b \)

Since the relationship is linear and the line passes through the origin (since \( y = 0 \) when \( x = 0 \)), the y-intercept \( b \) is 0.

Thus, the equation is:

\[
y = \frac{1}{35}x
\]

This equation represents the relationship between the number of miles traveled and the gallons of gas used.

Would you like any further details or have questions?

### Related Questions:
1. How can you verify that the equation \( y = \frac{1}{35}x \) is correct using other points from the table?
2. How would the equation change if the y-intercept were not 0?
3. What does the slope of \( \frac{1}{35} \) represent in the context of this problem?
4. Can you graph this equation and explain its significance?
5. How would the equation change if the gas efficiency changed to 40 miles per gallon?

### Tip:
When analyzing relationships from tables, always check if the change in \( x \) and \( y \) is consistent, which often indicates a linear function.

The table shows the relationship between the number of miles traveled, x, and the gallons of gas used, y. Write an equation to represent this relationship.

Learn how to write a linear equation representing the relationship between miles traveled and gallons of gas used. Using a step-by-step approach, this problem covers slope calculation and equation derivation. Suitable for students in Grades 7-9.

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