To find the equation of a line passing through the point \(A(3,3)\) with a given slope \(m = 3\), we can use the point-slope form of the equation of a line:

\[ y - y_1 = m(x - x_1) \]

Given:
- \(A(3,3)\)
- \(m = 3\)

Substitute the point \((x_1, y_1) = (3, 3)\) and the slope \(m = 3\) into the point-slope form:

\[ y - 3 = 3(x - 3) \]
\[ y - 3 = 3x - 9 \]
\[ y = 3x - 9 + 3 \]
\[ y = 3x - 6 \]

However, you mentioned \(y = 3x + 4\). Let's analyze the situation:
- If the line passes through \(A(3, 3)\) and has a slope of 3, the correct equation is \(y = 3x - 6\).

To confirm, substitute \(x = 3\) into \(y = 3x + 4\) to see if it passes through \(A(3, 3)\):

\[ y = 3(3) + 4 \]
\[ y = 9 + 4 \]
\[ y = 13 \]

Since \(y = 13\) does not equal \(3\), the line \(y = 3x + 4\) does not pass through the point \(A(3, 3)\).

### Final Equation
The correct equation of the line passing through the point \(A(3, 3)\) with a slope of 3 is:
\[ y = 3x - 6 \]

Would you like any further details or have any questions about these calculations?

Here are 8 related questions for deeper understanding:
1. How do you verify if a point lies on a given line?
2. What is the difference between the point-slope form and the slope-intercept form of a line equation?
3. How do you derive the slope-intercept form from the point-slope form?
4. What are the steps to find the equation of a line given two points?
5. How do you find the x- and y-intercepts of a line given its equation?
6. How does changing the slope affect the angle of a line on a graph?
7. What are some common mistakes to avoid when finding the equation of a line?
8. How can you use the equation of a line to find the distance from a point to the line?

### Tip:
Always double-check your substitution when verifying if a point lies on a line. This helps ensure the point satisfies the line's equation and avoids potential errors.

Learn how to find the equation of a line passing through point (3,3) with a slope of 3. This problem covers the use of the point-slope form of a line equation and verification of line equations using specific points. Suitable for students in Grades 7-9.

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