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We are asked to find the distance between two pairs of parallel lines in questions 23 and 24.

### For Question 23:
The equations of the lines are:

1. \( y = x - 6 \)
2. \( y = x + 8 \)

Both lines have the same slope (1), indicating they are parallel.

To calculate the distance between parallel lines of the form \( y = mx + b_1 \) and \( y = mx + b_2 \), we use the formula for the distance between two parallel lines:
\[
\text{Distance} = \frac{|b_2 - b_1|}{\sqrt{1+m^2}}
\]
Here, the slope \( m = 1 \), and the intercepts are \( b_1 = -6 \) and \( b_2 = 8 \).

Let's calculate the distance:

\[
\text{Distance} = \frac{|8 - (-6)|}{\sqrt{1+1^2}} = \frac{|8 + 6|}{\sqrt{2}} = \frac{14}{\sqrt{2}} = \frac{14}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{14\sqrt{2}}{2} = 7\sqrt{2}
\]

So, the distance between the lines in Question 23 is \( 7\sqrt{2} \).

### For Question 24:
The equations of the lines are:

1. \( y = -2x + 10 \)
2. \( y = -2x - 5 \)

Again, the lines are parallel since they have the same slope of -2.

Using the same formula as before:
\[
\text{Distance} = \frac{|b_2 - b_1|}{\sqrt{1+m^2}}
\]
Here, \( m = -2 \), and the intercepts are \( b_1 = 10 \) and \( b_2 = -5 \).

Let’s calculate:

\[
\text{Distance} = \frac{| -5 - 10 |}{\sqrt{1 + (-2)^2}} = \frac{| -15 |}{\sqrt{1 + 4}} = \frac{15}{\sqrt{5}} = \frac{15}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{15\sqrt{5}}{5} = 3\sqrt{5}
\]

Thus, the distance between the lines in Question 24 is \( 3\sqrt{5} \).

Let me know if you want more details about these calculations! Here are some related questions:

1. What is the formula for finding the distance between any two points?
2. How can we determine if two lines are parallel or perpendicular?
3. What are the implications of the slope in determining the orientation of a line?
4. How do you find the distance between two non-parallel lines?
5. What is the geometric meaning of the distance between parallel lines?

**Tip:** When working with parallel lines, remember that the slope stays constant, and only the intercepts vary, simplifying calculations.

Find the distance between each pair of parallel lines: y = x - 6 and y = x + 8; y = -2x + 10 and y = -2x - 5.

Learn how to calculate the distance between two parallel lines such as y = x - 6 and y = x + 8, using the formula |b2 - b1| / sqrt(1 + m^2). This step-by-step solution applies key algebraic concepts and geometric formulas, suitable for students in Grades 9-11.

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