To find the equation of a line parallel to $y = \frac{1}{3}x + 1$ that passes through the point $(5, -2)$, follow these steps:

Step 1: Identify the slope

The given line $y = \frac{1}{3}x + 1$ has a slope of $\frac{1}{3}$. A line parallel to this line will have the same slope: $m = \frac{1}{3}$.

Step 2: Use the point-slope formula

The equation of a line can be written using the point-slope form: $y - y_1 = m(x - x_1)$ Here:

• $m = \frac{1}{3}$ (slope)
• $(x_1, y_1) = (5, -2)$ (point)

Substitute these values: $y - (-2) = \frac{1}{3}(x - 5)$

Step 3: Simplify

$y + 2 = \frac{1}{3}x - \frac{5}{3}$ $y = \frac{1}{3}x - \frac{5}{3} - 2$ $y = \frac{1}{3}x - \frac{5}{3} - \frac{6}{3}$ $y = \frac{1}{3}x - \frac{11}{3}$

Final Answer:

The equation of the line is: $y = \frac{1}{3}x - \frac{11}{3}$

Let me know if you'd like a graph or more details!

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1. What does it mean for two lines to be parallel?
2. How would the equation change if the line passed through a different point?
3. What is the difference between the point-slope and slope-intercept forms of a line?
4. How do you determine if two lines are perpendicular?
5. Could you rewrite the final equation in standard form?

Tip: Always double-check signs when simplifying fractions in slope-intercept form!