Write The Equation Of The Line That Is Parallel To Y = 1 2 X − 8 Y=\frac{1}{2}x-8 Y = 2 1 ​ X − 8 And Passes Through The Point { (4,10)$}$.

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Introduction

In mathematics, particularly in the realm of geometry and algebra, parallel lines play a crucial role in understanding various concepts and theorems. A parallel line is a line that lies in the same plane as another given line but never intersects it, no matter how far they are extended. In this article, we will delve into the process of finding the equation of a line that is parallel to a given line and passes through a specific point.

Understanding the Given Line

The given line is represented by the equation y=12x−8y=\frac{1}{2}x-8. This is a linear equation in the slope-intercept form, where the slope is 12\frac{1}{2} and the y-intercept is −8-8. The slope of a line is a measure of how steep it is, and in this case, the slope is positive, indicating that the line slopes upward from left to right.

Finding the Slope of the Parallel Line

Since the line we are looking for is parallel to the given line, it must have the same slope as the given line. Therefore, the slope of the parallel line is also 12\frac{1}{2}.

Using the Point-Slope Form

To find the equation of the parallel line that passes through the point (4,10)(4,10), we can use the point-slope form of a linear equation, which is given by:

y−y1=m(x−x1)y-y_1=m(x-x_1)

where (x1,y1)(x_1,y_1) is the given point, and mm is the slope of the line.

Substituting the Values

Substituting the values of the given point (4,10)(4,10) and the slope m=12m=\frac{1}{2} into the point-slope form, we get:

y−10=12(x−4)y-10=\frac{1}{2}(x-4)

Simplifying the Equation

To simplify the equation, we can start by distributing the slope to the terms inside the parentheses:

y−10=12x−2y-10=\frac{1}{2}x-2

Next, we can add 1010 to both sides of the equation to isolate the term with the variable yy:

y=12x−2+10y=\frac{1}{2}x-2+10

Simplifying the right-hand side of the equation, we get:

y=12x+8y=\frac{1}{2}x+8

Conclusion

In this article, we have demonstrated how to find the equation of a line that is parallel to a given line and passes through a specific point. By using the point-slope form of a linear equation and substituting the values of the given point and the slope, we were able to derive the equation of the parallel line. The resulting equation is y=12x+8y=\frac{1}{2}x+8, which represents a line with a slope of 12\frac{1}{2} and a y-intercept of 88.

Key Takeaways

  • The equation of a line that is parallel to a given line has the same slope as the given line.
  • The point-slope form of a linear equation can be used to find the equation of a line that passes through a specific point.
  • By substituting the values of the given point and the slope into the point-slope form, we can derive the equation of the parallel line.

Real-World Applications

The concept of parallel lines has numerous real-world applications in various fields, including:

  • Architecture: Parallel lines are used in the design of buildings, bridges, and other structures to create a sense of balance and harmony.
  • Art: Parallel lines are used in various art forms, such as painting and sculpture, to create a sense of depth and perspective.
  • Engineering: Parallel lines are used in the design of machines and mechanisms to create a sense of efficiency and precision.

Final Thoughts

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

A: A parallel line is a line that lies in the same plane as another given line but never intersects it, no matter how far they are extended. A perpendicular line, on the other hand, is a line that intersects another line at a 90-degree angle.

Q: How do I determine if two lines are parallel or not?

A: To determine if two lines are parallel or not, you can use the following methods:

  • Check if the lines have the same slope. If they do, then they are parallel.
  • Check if the lines intersect at any point. If they do not intersect, then they are parallel.
  • Use the slope-intercept form of a linear equation to find the equation of one line and then check if the other line is parallel to it.

Q: What is the equation of a line that is parallel to the line y=2x+3y=2x+3 and passes through the point (4,10)(4,10)?

A: To find the equation of a line that is parallel to the given line and passes through the point (4,10)(4,10), we can use the point-slope form of a linear equation. The slope of the given line is 22, so the slope of the parallel line is also 22. Substituting the values of the given point and the slope into the point-slope form, we get:

y−10=2(x−4)y-10=2(x-4)

Simplifying the equation, we get:

y=2x−8+10y=2x-8+10

y=2x+2y=2x+2

Q: What is the equation of a line that is parallel to the line y=−12x+5y=-\frac{1}{2}x+5 and passes through the point (3,7)(3,7)?

A: To find the equation of a line that is parallel to the given line and passes through the point (3,7)(3,7), we can use the point-slope form of a linear equation. The slope of the given line is −12-\frac{1}{2}, so the slope of the parallel line is also −12-\frac{1}{2}. Substituting the values of the given point and the slope into the point-slope form, we get:

y−7=−12(x−3)y-7=-\frac{1}{2}(x-3)

Simplifying the equation, we get:

y=−12x+32+7y=-\frac{1}{2}x+\frac{3}{2}+7

y=−12x+172y=-\frac{1}{2}x+\frac{17}{2}

Q: How do I find the equation of a line that is parallel to a given line and passes through a specific point if the given line is in the form y=mx+by=mx+b?

A: To find the equation of a line that is parallel to a given line and passes through a specific point if the given line is in the form y=mx+by=mx+b, you can follow these steps:

  • Identify the slope of the given line, which is mm.
  • Use the point-slope form of a linear equation to find the equation of the parallel line.
  • Substitute the values of the given point and the slope into the point-slope form.
  • Simplify the equation to find the equation of the parallel line.

Q: What is the significance of parallel lines in real-world applications?

A: Parallel lines have numerous real-world applications in various fields, including:

  • Architecture: Parallel lines are used in the design of buildings, bridges, and other structures to create a sense of balance and harmony.
  • Art: Parallel lines are used in various art forms, such as painting and sculpture, to create a sense of depth and perspective.
  • Engineering: Parallel lines are used in the design of machines and mechanisms to create a sense of efficiency and precision.

Q: Can parallel lines be perpendicular to each other?

A: No, parallel lines cannot be perpendicular to each other. By definition, parallel lines are lines that lie in the same plane and never intersect, while perpendicular lines are lines that intersect at a 90-degree angle.