Let's solve this step-by-step:

We are given the equation of line $g$: $y = \frac{1}{3}x - 1$ We need to find the equation of line $h$, which is parallel to line $g$ and passes through the point $(10, 2)$.

1. Identify the slope of the line $h$:

Since line $h$ is parallel to line $g$, it will have the same slope. The equation of line $g$ is in slope-intercept form, $y = mx + b$, where $m$ is the slope. Here, the slope $m$ of line $g$ is $\frac{1}{3}$.

Thus, the slope of line $h$ is also $\frac{1}{3}$.

2. Use the point-slope form of the equation of a line:

The point-slope form of a line's equation is: $y - y_1 = m(x - x_1)$ where $m$ is the slope, and $(x_1, y_1)$ is a point on the line. For line $h$, we have:

• Slope $m = \frac{1}{3}$,
• Point $(x_1, y_1) = (10, 2)$.

Substituting these values into the point-slope form: $y - 2 = \frac{1}{3}(x - 10)$

3. Simplify to slope-intercept form:

Now, let's simplify this to get the equation in slope-intercept form: $y - 2 = \frac{1}{3}x - \frac{10}{3}$ Add 2 to both sides to solve for $y$: $y = \frac{1}{3}x - \frac{10}{3} + 2$ Convert 2 to a fraction with the same denominator: $y = \frac{1}{3}x - \frac{10}{3} + \frac{6}{3}$ Now, combine the constants: $y = \frac{1}{3}x - \frac{4}{3}$

Thus, the equation of line $h$ is: $y = \frac{1}{3}x - \frac{4}{3}$

Would you like further clarification on any step, or do you have additional questions?

Here are 5 related questions to expand your understanding:

1. How do you find the equation of a line that is perpendicular instead of parallel?
2. What happens to the slope when two lines are perpendicular?
3. How can you determine the equation of a line given two points?
4. Can you find the x- and y-intercepts of line $h$?
5. How would the equation change if line $h$ passed through a different point?

Tip: Always simplify fractions when combining constants, and make sure to adjust the denominator to match when adding or subtracting fractions.