To find the equation of the line through $(-2,1)$ that is parallel to $4x - 5y = 6$, follow these steps:

Step 1: Rewrite the given equation in slope-intercept form ($y = mx + b$):

Start with: $4x - 5y = 6$

Solve for $y$: $-5y = -4x + 6$ $y = \frac{4}{5}x - \frac{6}{5}$

The slope ($m$) of the line is $\frac{4}{5}$.

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

The formula is: $y - y_1 = m(x - x_1)$ Here, $(x_1, y_1) = (-2, 1)$ and $m = \frac{4}{5}$.

Substitute these values: $y - 1 = \frac{4}{5}(x - (-2))$ $y - 1 = \frac{4}{5}(x + 2)$

Step 3: Simplify the equation:

Distribute $\frac{4}{5}$: $y - 1 = \frac{4}{5}x + \frac{8}{5}$

Add 1 ($\frac{5}{5}$) to both sides: $y = \frac{4}{5}x + \frac{8}{5} + \frac{5}{5}$ $y = \frac{4}{5}x + \frac{13}{5}$

Final Answer:

The equation of the line is: $y = \frac{4}{5}x + \frac{13}{5}$

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Do you want further details or have any questions? Here are 5 related questions to explore:

1. How do you find the equation of a line perpendicular to $4x - 5y = 6$ through the same point?
2. What is the slope-intercept form of a line?
3. Can you find the x- and y-intercepts of the new line $y = \frac{4}{5}x + \frac{13}{5}$?
4. How would you write the equation of the parallel line in standard form?
5. What are the properties of parallel lines in terms of slopes?

Tip: Always remember that parallel lines have the same slope!