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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.

Find an equation of the line through (4,5) and parallel to 8x - 2y = 1

Learn how to find the equation of a line passing through the point (4,5) and parallel to the line 8x - 2y = 1. This step-by-step guide explores key algebraic concepts like slope-intercept form and point-slope form. Ideal for students in Grades 8-10.

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