Distance between Y equals 2X +3 and Y equals 2X +7

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

\[
y = 2x + 3 \quad \text{and} \quad y = 2x + 7,
\]

we use the formula for the distance between two parallel lines of the form \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\):

\[
\text{Distance} = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}}.
\]

Here, the lines are both in the slope-intercept form \(y = mx + b\), which can be rewritten in standard form \(Ax + By + C = 0\):

1. The first line \(y = 2x + 3\) becomes \(2x - y + 3 = 0\), so \(A = 2\), \(B = -1\), \(C_1 = 3\).
2. The second line \(y = 2x + 7\) becomes \(2x - y + 7 = 0\), so \(A = 2\), \(B = -1\), \(C_2 = 7\).

Now, applying the formula:

\[
\text{Distance} = \frac{|7 - 3|}{\sqrt{2^2 + (-1)^2}} = \frac{4}{\sqrt{4 + 1}} = \frac{4}{\sqrt{5}}.
\]

To express this distance in a simpler form, multiply by \(\frac{\sqrt{5}}{\sqrt{5}}\):

\[
\text{Distance} = \frac{4 \sqrt{5}}{5}.
\]

Thus, the distance between the lines \(y = 2x + 3\) and \(y = 2x + 7\) is \(\frac{4 \sqrt{5}}{5}\) units.

Would you like more details on this calculation or have any questions? Here are some related questions you might find interesting:

1. How do you find the distance between two points on a coordinate plane?
2. What is the formula for the distance between a point and a line?
3. How do you find the perpendicular distance from a point to a line?
4. Can we find the intersection point between two non-parallel lines?
5. How do you find the angle between two lines?

**Tip:** For parallel lines, the distance between them is constant and depends only on their perpendicular separation!

Learn how to find the distance between two parallel lines, Y = 2X + 3 and Y = 2X + 7, using a step-by-step application of the distance formula for parallel lines. Suitable for students in Grades 9-12, this problem covers key algebraic and geometric concepts.

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