Line ℓ1 has the equation y=–
2x–2 and line ℓ2 has the equation y=–
2x+2. Find the distance between ℓ1 and ℓ2.
Round your answer to the nearest tenth.

To find the distance between the two parallel lines \( \ell_1 \) and \( \ell_2 \), given their equations:

\[
\ell_1: y = -2x - 2
\]
\[
\ell_2: y = -2x + 2
\]

We use the formula for the distance \( d \) between two parallel lines of the form \( y = mx + c_1 \) and \( y = mx + c_2 \):

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

Here, \( m = -2 \), \( c_1 = -2 \), and \( c_2 = 2 \). Plugging these values into the formula, we get:

\[
d = \frac{|2 - (-2)|}{\sqrt{1 + (-2)^2}} = \frac{|2 + 2|}{\sqrt{1 + 4}} = \frac{4}{\sqrt{5}}
\]

Now, let's compute \( \frac{4}{\sqrt{5}} \):

\[
d = \frac{4}{\sqrt{5}} \approx \frac{4}{2.236} \approx 1.788
\]

Rounding to the nearest tenth:

\[
d \approx 1.8
\]

So, the distance between the lines \( \ell_1 \) and \( \ell_2 \) is approximately **1.8 units**.

Would you like more details or have any questions?

Here are some related questions:

1. How do you find the distance between a point and a line?
2. What is the geometric interpretation of parallel lines in the coordinate plane?
3. Can the distance between non-parallel lines be defined similarly?
4. How does the slope of a line affect its distance to another parallel line?
5. What changes in the formula if the lines have different slopes?

**Tip:** When dealing with parallel lines, always remember their slopes are identical, which simplifies many calculations.

Discover how to find the distance between two parallel lines y = -2x - 2 and y = -2x + 2 using the distance formula. This problem involves calculating the distance between parallel lines given their equations. Suitable for Grades 9-12.

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