Understanding the Given Linear Equation

Oct 13, 2025 by recoveryshake.com 45 views

Introduction

In mathematics, linear functions are a fundamental concept in algebra and geometry. A linear function is a polynomial function of degree one, and it can be represented in the form of g(x) = mx + b, where m is the slope and b is the y-intercept. In this article, we will find an equation for the linear function g(x) which is perpendicular to the linear 5x-4y=8 and intersects the line 5x-4y=8 at x=20.

Understanding the Given Linear Equation

The given linear equation is 5x-4y=8. To find the slope of this line, we need to rewrite the equation in the slope-intercept form, which is y = mx + b. To do this, we can isolate y on one side of the equation.

5x - 4y = 8
-4y = -5x + 8
y = (5/4)x - 2

From the above equation, we can see that the slope of the line is 5/4.

Finding the Slope of the Perpendicular Line

Since the line g(x) is perpendicular to the given line, the slope of g(x) will be the negative reciprocal of the slope of the given line. The negative reciprocal of 5/4 is -4/5.

Finding the Equation of the Perpendicular Line

Now that we have the slope of the perpendicular line, we can use the point-slope form of a linear equation to find the equation of the line. 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.

We are given that the line intersects the line 5x-4y=8 at x=20. To find the corresponding y-coordinate, we can substitute x=20 into the equation 5x-4y=8.

5(20) - 4y = 8
100 - 4y = 8
-4y = -92
y = 23

So, the point of intersection is (20, 23).

Now, we can use the point-slope form to find the equation of the perpendicular line.

y - 23 = (-4/5)(x - 20)
y - 23 = (-4/5)x + 16
y = (-4/5)x + 39

Conclusion

In this article, we found an equation for the linear function g(x) which is perpendicular to the linear 5x-4y=8 and intersects the line 5x-4y=8 at x=20. The equation of the perpendicular line is y = (-4/5)x + 39.

Discussion

The concept of perpendicular lines is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, and computer science. The ability to find the equation of a perpendicular line is an essential skill that can be used to solve a wide range of problems.

Example Use Cases

Physics: In physics, the concept of perpendicular lines is used to describe the motion of objects. For example, the trajectory of a projectile can be described using the equation of a perpendicular line.
Engineering: In engineering, the concept of perpendicular lines is used to design and build structures such as bridges and buildings. The equation of a perpendicular line can be used to ensure that the structure is stable and secure.
Computer Science: In computer science, the concept of perpendicular lines is used in computer graphics and game development. The equation of a perpendicular line can be used to create realistic and immersive graphics.

Final Thoughts

In conclusion, finding the equation of a perpendicular line is an essential skill that can be used to solve a wide range of problems in various fields. The concept of perpendicular lines is a fundamental concept in mathematics, and it has numerous applications in various fields. By understanding the concept of perpendicular lines, we can develop a deeper understanding of the world around us and solve complex problems with ease.

Introduction

In our previous article, we found an equation for the linear function g(x) which is perpendicular to the linear 5x-4y=8 and intersects the line 5x-4y=8 at x=20. In this article, we will answer some frequently asked questions related to the concept of perpendicular lines and the equation of a linear function.

Q&A

Q1: What is the concept of perpendicular lines?

A1: The concept of perpendicular lines refers to two lines that intersect at a 90-degree angle. In other words, the product of the slopes of two perpendicular lines is -1.

Q2: How do you find the equation of a perpendicular line?

A2: To find the equation of a perpendicular line, you need to know the slope of the original line and the point of intersection. You can use the point-slope form of a linear equation to find the equation of the perpendicular line.

Q3: What is the slope of the perpendicular line?

A3: The slope of the perpendicular line is the negative reciprocal of the slope of the original line. In the case of the line 5x-4y=8, the slope is 5/4, so the slope of the perpendicular line is -4/5.

Q4: How do you find the equation of a line that intersects another line at a given point?

A4: To find the equation of a line that intersects another line at a given point, you can use the point-slope form of a linear equation. You need to know the point of intersection and the slope of the line.

Q5: What is the equation of the perpendicular line that intersects the line 5x-4y=8 at x=20?

A5: The equation of the perpendicular line that intersects the line 5x-4y=8 at x=20 is y = (-4/5)x + 39.

Q6: How do you use the equation of a perpendicular line in real-world applications?

A6: The equation of a perpendicular line can be used in various real-world applications such as physics, engineering, and computer science. For example, the trajectory of a projectile can be described using the equation of a perpendicular line.

Q7: What are some common mistakes to avoid when finding the equation of a perpendicular line?

A7: Some common mistakes to avoid when finding the equation of a perpendicular line include:

Not knowing the slope of the original line
Not knowing the point of intersection
Not using the point-slope form of a linear equation
Not checking the equation for accuracy

Q8: How do you check the accuracy of the equation of a perpendicular line?

A8: To check the accuracy of the equation of a perpendicular line, you can substitute the point of intersection into the equation and verify that it satisfies the equation.

Conclusion

In this article, we answered some frequently asked questions related to the concept of perpendicular lines and the equation of a linear function. We hope that this article has provided you with a better understanding of the concept of perpendicular lines and how to find the equation of a linear function.

Discussion

Example Use Cases

Physics: In physics, the concept of perpendicular lines is used to describe the motion of objects. For example, the trajectory of a projectile can be described using the equation of a perpendicular line.
Engineering: In engineering, the concept of perpendicular lines is used to design and build structures such as bridges and buildings. The equation of a perpendicular line can be used to ensure that the structure is stable and secure.
Computer Science: In computer science, the concept of perpendicular lines is used in computer graphics and game development. The equation of a perpendicular line can be used to create realistic and immersive graphics.