We are asked to find the equation of a line that passes through the point $(4, 5)$ and is parallel to the line given by $8x - 2y = 1$.

Step 1: Determine the slope of the given line

We can first find the slope of the line $8x - 2y = 1$ by rewriting it in slope-intercept form, $y = mx + b$, where $m$ is the slope.

$8x - 2y = 1$

Solve for $y$:

$-2y = -8x + 1$

$y = 4x - \frac{1}{2}$

From this equation, we can see that the slope $m$ of the given line is $4$.

Step 2: Use the slope-point form

Since the line we need to find is parallel to the given line, it will have the same slope. Therefore, the slope of the new line is also $4$.

Now, using the point-slope form of a line equation:

$y - y_1 = m(x - x_1)$

Where $m = 4$, $(x_1, y_1) = (4, 5)$. Substituting these values:

$y - 5 = 4(x - 4)$

Step 3: Simplify the equation

$y - 5 = 4x - 16$

$y = 4x - 16 + 5$

$y = 4x - 11$

Final Answer:

The equation of the line passing through $(4, 5)$ and parallel to $8x - 2y = 1$ is:

$y = 4x - 11$

Would you like further details or clarification?

Related Questions:

1. How do you find the slope of a line from its equation?
2. What is the general form of the point-slope equation of a line?
3. How can you verify if two lines are parallel?
4. How do you convert a linear equation from standard form to slope-intercept form?
5. Can a line parallel to another line ever intersect it?

Tip:

When two lines are parallel, their slopes are always equal! Use this to your advantage when writing the equation of a line.