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To find the distance between the two parallel lines given by the equations:

- \( y = -3x + 1 \)
- \( y = -3x + 4 \)

We use the formula for the distance between two parallel lines in slope-intercept form:

\[
d = \frac{|c_2 - c_1|}{\sqrt{1 + m^2}}
\]

Where:
- \( m \) is the slope of the lines (which is the same for both lines since they are parallel),
- \( c_1 \) and \( c_2 \) are the y-intercepts of the two lines.

### Step 1: Identify the slope and y-intercepts
- The slope \( m = -3 \).
- For the line \( y = -3x + 1 \), \( c_1 = 1 \).
- For the line \( y = -3x + 4 \), \( c_2 = 4 \).

### Step 2: Plug the values into the distance formula
\[
d = \frac{|4 - 1|}{\sqrt{1 + (-3)^2}}
\]
\[
d = \frac{|3|}{\sqrt{1 + 9}} = \frac{3}{\sqrt{10}} = \frac{3}{3.162}
\]

### Step 3: Calculate the distance
\[
d \approx 0.9487 \text{ units}.
\]

Rounding to the nearest tenth, the distance between the parallel lines is approximately **0.9 units**.

Would you like further clarification on any part of this, or do you have any other questions?

Here are some related questions you might find useful:
1. How do you find the slope of a line from its equation?
2. How do you find the distance between two points on a coordinate plane?
3. Can you calculate the distance between lines that are not parallel?
4. How does changing the slope affect the distance between lines?
5. What is the geometric significance of parallel lines having the same slope?

**Tip:** Always double-check the slope and y-intercepts when using the distance formula for parallel lines.

Find the distance between the parallel lines. If necessary, round your answer to the nearest tenth.

Learn how to calculate the distance between two parallel lines with the equations y = -3x + 1 and y = -3x + 4. This guide provides a step-by-step solution using the distance formula for parallel lines, ideal for high school students studying geometry or algebra.

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