To find the equation of the line that passes through the point $(1, 2)$ and is parallel to the line $2x - 5y = 8$, we follow these steps:

Step 1: Find the slope of the given line.

The equation of the line is given in the standard form: $2x - 5y = 8$ To find the slope, $m$, we first rewrite the equation in slope-intercept form $y = mx + b$, where $m$ is the slope.

Starting with: $2x - 5y = 8$

Subtract $2x$ from both sides: $-5y = -2x + 8$

Divide every term by $-5$: $y = \frac{2}{5}x - \frac{8}{5}$

So, the slope $m$ of the given line is: $m = \frac{2}{5}$

Step 2: Use the slope-point form to write the equation of the parallel line.

Since parallel lines have the same slope, the line passing through $(1, 2)$ will also have a slope of $\frac{2}{5}$.

The point-slope form of a line is given by: $y - y_1 = m(x - x_1)$ Substitute $m = \frac{2}{5}$, $x_1 = 1$, and $y_1 = 2$: $y - 2 = \frac{2}{5}(x - 1)$

Step 3: Simplify the equation.

Distribute the slope $\frac{2}{5}$: $y - 2 = \frac{2}{5}x - \frac{2}{5}$

Add 2 to both sides to solve for $y$: $y = \frac{2}{5}x - \frac{2}{5} + \frac{10}{5}$ $y = \frac{2}{5}x + \frac{8}{5}$

Thus, the equation of the line passing through $(1, 2)$ and parallel to $2x - 5y = 8$ is: $y = \frac{2}{5}x + \frac{8}{5}$

Would you like any further details or have any questions?

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Here are some related questions:

1. How do you find the equation of a line perpendicular to a given line?
2. What are the properties of parallel lines in the coordinate plane?
3. How can the point-slope form be converted to the standard form of a line equation?
4. What is the significance of the slope-intercept form in graphing lines?
5. How do you determine if two lines are parallel just by looking at their equations?

Tip: Always convert the given line equation to slope-intercept form to easily identify the slope, which is crucial for finding parallel and perpendicular lines.