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To determine which of the given lines is parallel to the line \( y = 4x + 1 \) and passes through the point \( (5, 7) \), we need to break down the problem step by step.

### Step 1: Identify the slope of the given line.
The equation of the line \( y = 4x + 1 \) is in slope-intercept form \( y = mx + b \), where \( m \) represents the slope. So, the slope of this line is \( m = 4 \).

### Step 2: Recognize parallel lines.
Two lines are parallel if and only if their slopes are equal. This means that we are looking for a line among the given options with the same slope, \( m = 4 \).

### Step 3: Convert each equation to slope-intercept form.
We will convert each of the given equations from the standard form to slope-intercept form to identify their slopes.

1. **Equation 1:** \( 4x - y - 23 = 0 \)
   \[
   4x - y = 23 \quad \Rightarrow \quad y = 4x - 23
   \]
   The slope here is \( m = 4 \).

2. **Equation 2:** \( 4x - y - 13 = 0 \)
   \[
   4x - y = 13 \quad \Rightarrow \quad y = 4x - 13
   \]
   The slope here is also \( m = 4 \).

3. **Equation 3:** \( 4x + y + 13 = 0 \)
   \[
   4x + y = -13 \quad \Rightarrow \quad y = -4x - 13
   \]
   The slope here is \( m = -4 \).

### Step 4: Check the point \( (5, 7) \).
Now that we know both Equation 1 and Equation 2 have slopes of \( 4 \), we need to check which one passes through the point \( (5, 7) \).

- **For Equation 1:** \( y = 4x - 23 \)
  Substituting \( x = 5 \):
  \[
  y = 4(5) - 23 = 20 - 23 = -3
  \]
  This does not match \( y = 7 \), so this is not the correct line.

- **For Equation 2:** \( y = 4x - 13 \)
  Substituting \( x = 5 \):
  \[
  y = 4(5) - 13 = 20 - 13 = 7
  \]
  This matches \( y = 7 \), so **Equation 2 is the correct line**.

### Final Answer:
The correct equation of the line that is parallel to \( y = 4x + 1 \) and passes through \( (5, 7) \) is:
\[
\boxed{4x - y - 13 = 0}
\]

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

1. What is the general form of a linear equation and how can it be converted to slope-intercept form?
2. How do you find the slope of a line given two points on the line?
3. Why do parallel lines have the same slope?
4. How can we determine if two lines are perpendicular?
5. What is the relationship between the slope of a line and its angle of inclination?

**Tip:** When dealing with lines, always check for parallelism by comparing slopes.

Which of the following lines is parallel to the line y = 4x + 1 and passes through the point (5, 7)?

Discover how to identify a line parallel to y = 4x + 1 that passes through the point (5, 7). This detailed explanation breaks down key algebraic concepts like slope and linear equations, with step-by-step solutions for students in Grades 8-10.

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