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Let's solve each problem step by step.

### Problem 9
We are given:
- Point: \( (3, 3) \)
- Slope: \( m = -\frac{1}{6} \)

We want to find the equation of the line in standard form. The point-slope form of a line is:

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

Substitute the given point and slope:

\[
y - 3 = -\frac{1}{6}(x - 3)
\]

Simplify:

\[
y - 3 = -\frac{1}{6}x + \frac{1}{2}
\]

Now add 3 to both sides:

\[
y = -\frac{1}{6}x + \frac{1}{2} + 3
\]

Convert 3 to a fraction:

\[
y = -\frac{1}{6}x + \frac{1}{2} + \frac{6}{2} = -\frac{1}{6}x + \frac{7}{2}
\]

To write this in standard form, we eliminate fractions by multiplying the entire equation by 6:

\[
6y = -x + 21
\]

Now rearrange to standard form \(Ax + By = C\):

\[
x + 6y = 21
\]

Thus, the equation in standard form is:

\[
x + 6y = 21
\]

---

### Problem 10
We are given:
- Point: \( (2, -5) \)
- Slope: \( m = -\frac{5}{2} \)

Using the point-slope form again:

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

Substitute the given point and slope:

\[
y + 5 = -\frac{5}{2}(x - 2)
\]

Simplify:

\[
y + 5 = -\frac{5}{2}x + 5
\]

Subtract 5 from both sides:

\[
y = -\frac{5}{2}x
\]

To eliminate the fraction, multiply the entire equation by 2:

\[
2y = -5x
\]

Rearranging to standard form:

\[
5x + 2y = 0
\]

Thus, the equation in standard form is:

\[
5x + 2y = 0
\]

---

Let me know if you would like further details on the solutions.

### Follow-up questions:
1. How is point-slope form derived from the slope formula?
2. What steps are needed to convert slope-intercept form into standard form?
3. What is the advantage of using the standard form of a line equation?
4. How would you graph a line using its standard form equation?
5. What does it mean for coefficients in the standard form to be integral?

### Tip:
When converting to standard form, make sure the coefficient of \(x\) is positive and all coefficients are integers!

Write the standard form of the equation of the line with integral coefficients through the given point with the given slope.

Learn how to write the standard form of the equation of a line passing through a given point and having a specific slope. This problem covers key algebra concepts, such as point-slope form, and how to transform it into standard form. Suitable for Grades 9-12.

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