find the distance between the lines y = 1/2x -1 and y = 1/2x plus 6

The lines you're working with are:

1. \( y = \frac{1}{2}x - 1 \)
2. \( y = \frac{1}{2}x + 6 \)

Since both lines have the same slope (\( \frac{1}{2} \)), they are parallel. To find the distance between two parallel lines, we use the formula:

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

Where:
- \( m \) is the slope of the lines,
- \( c_1 \) and \( c_2 \) are the y-intercepts of the lines.

For your lines:
- The slope \( m = \frac{1}{2} \),
- \( c_1 = -1 \) (from the first equation),
- \( c_2 = 6 \) (from the second equation).

Now, we plug these into the formula:

\[
d = \frac{|6 - (-1)|}{\sqrt{1 + \left(\frac{1}{2}\right)^2}} = \frac{|6 + 1|}{\sqrt{1 + \frac{1}{4}}} = \frac{7}{\sqrt{\frac{5}{4}}} = \frac{7}{\frac{\sqrt{5}}{2}} = \frac{14}{\sqrt{5}}.
\]

To rationalize the denominator, we multiply by \( \frac{\sqrt{5}}{\sqrt{5}} \):

\[
d = \frac{14\sqrt{5}}{5}.
\]

Thus, the distance between the two parallel lines is \( \frac{14\sqrt{5}}{5} \).

Would you like a more detailed explanation of any of these steps? 

Here are 5 related questions for further practice:
1. What is the formula for the distance between two parallel lines with different slopes?
2. How do you find the slope of a line given two points on the line?
3. How would the calculation change if the lines had different slopes?
4. How do you find the intersection point of two non-parallel lines?
5. What is the shortest distance between a point and a line?

**Tip:** Always ensure both lines are in the same format (slope-intercept form) before calculating the distance.

Find the distance between the lines y = 1/2x - 1 and y = 1/2x + 6

Calculate the distance between two parallel lines y = 1/2x - 1 and y = 1/2x + 6 using the formula for the distance between parallel lines. This problem involves algebraic and geometric concepts suitable for Grades 9-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation