In The { Xy$}$-plane, Line { \ell$}$ Passes Through The Point { (0,0)$}$ And Is Parallel To The Line Represented By The Equation { Y=8x+2$}$. If Line { \ell$}$ Also Passes Through The Point [$(2,

by ADMIN 196 views

Introduction

In the xy-plane, lines are represented by equations that describe their slope and y-intercept. When two lines are parallel, they have the same slope but different y-intercepts. In this article, we will explore how to find the equation of a line that passes through a given point and is parallel to another line represented by an equation.

Understanding Line Equations

A line in the xy-plane can be represented by the equation y = mx + b, where m is the slope of the line and b is the y-intercept. The slope of a line is a measure of how steep it is, and it can be calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

Parallel Lines

Two lines are parallel if they have the same slope but different y-intercepts. This means that if we have two lines represented by the equations y = mx + b1 and y = mx + b2, where m is the same for both lines, then the lines are parallel.

Given Information

In this problem, we are given that line â„“ passes through the point (0,0) and is parallel to the line represented by the equation y = 8x + 2. We are also given that line â„“ passes through the point (2, k), where k is an unknown value.

Finding the Equation of Line â„“

Since line â„“ is parallel to the line represented by the equation y = 8x + 2, it must have the same slope, which is 8. We can use the point-slope form of a line to find the equation of line â„“. The point-slope form is given by y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.

Using the Point-Slope Form

We can use the point-slope form to find the equation of line â„“ by substituting the given point (0,0) and the slope 8 into the equation. This gives us:

y - 0 = 8(x - 0)

Simplifying the equation, we get:

y = 8x

However, this is not the complete equation of line â„“, as it does not take into account the point (2, k). To find the complete equation, we need to use the point (2, k) and the slope 8 to find the y-intercept.

Finding the Y-Intercept

We can use the point (2, k) and the slope 8 to find the y-intercept by substituting these values into the equation y = 8x + b. This gives us:

k = 8(2) + b

Simplifying the equation, we get:

k = 16 + b

Solving for b

We can solve for b by rearranging the equation k = 16 + b to isolate b. This gives us:

b = k - 16

Finding the Equation of Line â„“

Now that we have found the y-intercept b, we can find the complete equation of line â„“ by substituting this value into the equation y = 8x + b. This gives us:

y = 8x + (k - 16)

Simplifying the equation, we get:

y = 8x + k - 16

Conclusion

In this article, we have explored how to find the equation of a line that passes through a given point and is parallel to another line represented by an equation. We have used the point-slope form of a line and the given information to find the equation of line â„“. The final equation of line â„“ is y = 8x + k - 16.

Final Answer

The final answer is y = 8x + k - 16.

Discussion

This problem is a classic example of how to find the equation of a line that passes through a given point and is parallel to another line represented by an equation. The key concept is to use the point-slope form of a line and the given information to find the equation of the line.

Related Problems

  • Find the equation of a line that passes through the point (1,2) and is parallel to the line represented by the equation y = 3x - 2.
  • Find the equation of a line that passes through the point (0,0) and is parallel to the line represented by the equation y = 2x + 1.

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak

Keywords

  • Line equation
  • Parallel lines
  • Point-slope form
  • Slope
  • Y-intercept
  • Equation of a line
    Q&A: Line Equation and Parallel Lines =====================================

Q: What is the equation of a line that passes through the point (0,0) and is parallel to the line represented by the equation y = 8x + 2?

A: The equation of the line is y = 8x + b, where b is the y-intercept. To find the value of b, we can use the point (2, k) and the slope 8 to find the y-intercept.

Q: How do I find the y-intercept of a line?

A: To find the y-intercept of a line, you can use the point-slope form of a line and the given information. For example, if you have a line that passes through the point (2, k) and has a slope of 8, you can use the equation k = 8(2) + b to find the y-intercept.

Q: What is the difference between a line and a parallel line?

A: A line is a set of points that extend infinitely in two directions, while a parallel line is a line that has the same slope as another line but does not intersect it.

Q: How do I determine if two lines are parallel?

A: To determine if two lines are parallel, you can compare their slopes. If the slopes are equal, then the lines are parallel.

Q: What is the point-slope form of a line?

A: The point-slope form of a line is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.

Q: How do I use the point-slope form to find the equation of a line?

A: To use the point-slope form to find the equation of a line, you can substitute the given point and the slope into the equation. For example, if you have a line that passes through the point (0,0) and has a slope of 8, you can use the equation y - 0 = 8(x - 0) to find the equation of the line.

Q: What is the equation of a line that passes through the point (1,2) and is parallel to the line represented by the equation y = 3x - 2?

A: The equation of the line is y = 3x + b, where b is the y-intercept. To find the value of b, you can use the point (1,2) and the slope 3 to find the y-intercept.

Q: How do I find the equation of a line that passes through two points?

A: To find the equation of a line that passes through two points, you can use the point-slope form of a line and the given information. For example, if you have two points (x1, y1) and (x2, y2), you can use the equation y - y1 = m(x - x1) to find the equation of the line.

Q: What is the equation of a line that passes through the point (0,0) and is parallel to the line represented by the equation y = 2x + 1?

A: The equation of the line is y = 2x + b, where b is the y-intercept. To find the value of b, you can use the point (0,0) and the slope 2 to find the y-intercept.

Q: How do I use the equation of a line to find the slope and y-intercept?

A: To use the equation of a line to find the slope and y-intercept, you can rearrange the equation to isolate the slope and y-intercept. For example, if you have the equation y = 8x + b, you can rearrange it to find the slope (m = 8) and y-intercept (b = k - 16).

Q: What is the significance of the y-intercept in a line?

A: The y-intercept is the point where the line intersects the y-axis. It is an important concept in mathematics and is used to find the equation of a line.

Q: How do I use the y-intercept to find the equation of a line?

A: To use the y-intercept to find the equation of a line, you can substitute the y-intercept into the equation of the line. For example, if you have a line that passes through the point (0,0) and has a y-intercept of b, you can use the equation y = 8x + b to find the equation of the line.

Q: What is the relationship between the slope and y-intercept of a line?

A: The slope and y-intercept of a line are related in that the slope is the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line, while the y-intercept is the point where the line intersects the y-axis.

Q: How do I use the relationship between the slope and y-intercept to find the equation of a line?

A: To use the relationship between the slope and y-intercept to find the equation of a line, you can substitute the slope and y-intercept into the equation of the line. For example, if you have a line that passes through the point (0,0) and has a slope of 8 and a y-intercept of b, you can use the equation y = 8x + b to find the equation of the line.

Q: What is the significance of the point-slope form of a line?

A: The point-slope form of a line is a powerful tool for finding the equation of a line. It allows you to use the slope and a point on the line to find the equation of the line.

Q: How do I use the point-slope form to find the equation of a line?

A: To use the point-slope form to find the equation of a line, you can substitute the given point and the slope into the equation. For example, if you have a line that passes through the point (0,0) and has a slope of 8, you can use the equation y - 0 = 8(x - 0) to find the equation of the line.

Q: What is the relationship between the point-slope form and the slope-intercept form of a line?

A: The point-slope form and the slope-intercept form of a line are related in that the point-slope form can be used to find the equation of a line, while the slope-intercept form can be used to find the slope and y-intercept of a line.

Q: How do I use the relationship between the point-slope form and the slope-intercept form to find the equation of a line?

A: To use the relationship between the point-slope form and the slope-intercept form to find the equation of a line, you can substitute the slope and y-intercept into the equation of the line. For example, if you have a line that passes through the point (0,0) and has a slope of 8 and a y-intercept of b, you can use the equation y = 8x + b to find the equation of the line.

Q: What is the significance of the slope-intercept form of a line?

A: The slope-intercept form of a line is a powerful tool for finding the equation of a line. It allows you to use the slope and y-intercept to find the equation of the line.

Q: How do I use the slope-intercept form to find the equation of a line?

A: To use the slope-intercept form to find the equation of a line, you can substitute the slope and y-intercept into the equation. For example, if you have a line that passes through the point (0,0) and has a slope of 8 and a y-intercept of b, you can use the equation y = 8x + b to find the equation of the line.

Q: What is the relationship between the slope-intercept form and the point-slope form of a line?

A: The slope-intercept form and the point-slope form of a line are related in that the slope-intercept form can be used to find the equation of a line, while the point-slope form can be used to find the slope and y-intercept of a line.

Q: How do I use the relationship between the slope-intercept form and the point-slope form to find the equation of a line?

A: To use the relationship between the slope-intercept form and the point-slope form to find the equation of a line, you can substitute the slope and y-intercept into the equation of the line. For example, if you have a line that passes through the point (0,0) and has a slope of 8 and a y-intercept of b, you can use the equation y = 8x + b to find the equation of the line.

Q: What is the significance of the equation of a line?

A: The equation of a line is a powerful tool for finding the slope and y-intercept of a line. It allows you to use the slope and y-intercept to find the equation of the line.

Q: How do I use the equation of a line to find the slope and y-intercept?

A: To use the equation of a line to