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The problem involves determining the correct linear equation based on the information given:

- The relationship between two variables, \(x\) and \(y\), is linear.
- For every increase in the value of \(x\) by 1, the value of \(y\) increases by 8. This tells us the slope is 8.
- When \(x = 2\), the value of \(y = 18\). This provides a point on the line: \((2, 18)\).

### Step-by-Step Solution:

The general form of a linear equation is:
\[
y = mx + b
\]
where \(m\) is the slope and \(b\) is the y-intercept.

1. **Determine the slope \(m\)**:
   From the problem, the slope \(m\) is given as 8 (since \(y\) increases by 8 when \(x\) increases by 1). So, the equation becomes:
   \[
   y = 8x + b
   \]

2. **Use the point \((x = 2, y = 18)\) to solve for \(b\)**:
   Substitute \(x = 2\) and \(y = 18\) into the equation:
   \[
   18 = 8(2) + b
   \]
   \[
   18 = 16 + b
   \]
   \[
   b = 18 - 16 = 2
   \]

Thus, the equation is:
\[
y = 8x + 2
\]

### Correct Answer:
The correct equation that represents this relationship is \(y = 8x + 2\), which corresponds to option **C**.

Would you like further details or have any questions?

Here are 5 related questions for further practice:
1. How do you find the slope of a line given two points?
2. What does the y-intercept of a linear equation represent in a graph?
3. How would you graph the equation \(y = 8x + 2\)?
4. How do changes in the slope affect the steepness of a line?
5. How do you interpret a point on the graph of a linear equation?

**Tip:** When solving linear equations, remember that the slope represents the rate of change and the y-intercept is the value of \(y\) when \(x = 0\).

The relationship between two variables, x and y, is linear. For every increase in the value of x by 1, the value of y increases by 8. When the value of x is 2, the value of y is 18. Which equation represents this relationship?

Learn how to determine the equation of a linear relationship where the slope is 8, and the point (2, 18) lies on the line. This problem covers the basics of slope-intercept form and solving for the y-intercept. Suitable for students in Grades 8-10.

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