In the xy-plane, line 
𝓁
 passes through the point (0,0)
 and is parallel to the line represented by the equation y=8x+2
. If line 
𝓁
 also passes through the point (3,d)
, what is the value of 
d?

To solve this problem, let's first understand the situation:

- Line \( \ell \) passes through the origin, which means its equation can be written as \( y = mx \), where \( m \) is the slope.
- Line \( \ell \) is also parallel to the line given by the equation \( y = 8x + 2 \).

Parallel lines have the same slope. Therefore, the slope \( m \) of line \( \ell \) is the same as the slope of the given line \( y = 8x + 2 \). The slope of this line is 8 (the coefficient of \( x \)).

So, the equation of line \( \ell \) is:
\[
y = 8x
\]

Since line \( \ell \) passes through the point \( (3, d) \), we can substitute \( x = 3 \) into the equation \( y = 8x \) to find the value of \( d \).

\[
d = 8(3) = 24
\]

Thus, the value of \( d \) is 24.

Would you like to explore more about this problem or have any other questions?

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**Relative Questions:**
1. How do you find the equation of a line that passes through two given points?
2. What would be the equation of line \( \ell \) if it passed through a different point other than the origin?
3. How do you determine if two lines are perpendicular?
4. What is the significance of the y-intercept in a line equation?
5. How do you convert the slope-intercept form of a line to the point-slope form?
6. How do you find the distance between a point and a line in the xy-plane?
7. How can you graph line \( \ell \) and the given line \( y = 8x + 2 \)?
8. What is the relationship between the slopes of two lines that are parallel?

**Tip:** When dealing with parallel lines, remember that their slopes are always equal, which simplifies finding the equation of a line given certain conditions.

Understand how to solve a problem involving lines and parallel lines in the xy-plane. This problem discusses the concept of parallel lines, their slopes, and finding a specific point on a line. Suitable for Grades 9-10 students.

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