The Simplified Cartesian Equation

Oct 13, 2025 by magicalceylontours.com 13 views

Introduction

In mathematics, particularly in the study of parametric equations, eliminating the parameter is a crucial step in finding a simplified Cartesian equation. This process involves expressing the parametric equations in terms of a single variable, usually x or y, and then eliminating the parameter to obtain a linear equation in the form of y = mx + b. In this article, we will explore how to eliminate the parameter to find a simplified Cartesian equation for a given set of parametric equations.

The Parametric Equations

The given parametric equations are:

x(t) = -7 - t y(t) = 5 - 3t

These equations represent the coordinates of a point in the Cartesian plane as a function of the parameter t.

Eliminating the Parameter

To eliminate the parameter, we need to express one of the variables in terms of the other. In this case, we can solve the first equation for t:

t = -x - 7

Now, substitute this expression for t into the second equation:

y(t) = 5 - 3t y(-x - 7) = 5 - 3(-x - 7)

Simplify the equation:

y(-x - 7) = 5 + 3x + 21 y = 3x + 26

The Simplified Cartesian Equation

The simplified Cartesian equation is:

y = 3x + 26

This equation represents a linear relationship between x and y, where the slope is 3 and the y-intercept is 26.

Discussion

Eliminating the parameter is an essential step in finding a simplified Cartesian equation. By expressing one of the variables in terms of the other, we can obtain a linear equation in the form of y = mx + b. This process is useful in various applications, such as graphing parametric equations, finding the equation of a tangent line, and solving systems of equations.

Example

Consider the parametric equations:

x(t) = 2t - 1 y(t) = t^2 + 1

To eliminate the parameter, we can solve the first equation for t:

t = (x + 1) / 2

Now, substitute this expression for t into the second equation:

y(t) = t^2 + 1 y((x + 1) / 2) = ((x + 1) / 2)^2 + 1

Simplify the equation:

y((x + 1) / 2) = (x^2 + 2x + 1) / 4 + 1 y = (x^2 + 2x + 5) / 4

The simplified Cartesian equation is:

y = (x^2 + 2x + 5) / 4

Conclusion

In conclusion, eliminating the parameter is a crucial step in finding a simplified Cartesian equation. By expressing one of the variables in terms of the other, we can obtain a linear equation in the form of y = mx + b. This process is useful in various applications, such as graphing parametric equations, finding the equation of a tangent line, and solving systems of equations.

References

[1] Anton, H. (2018). Calculus: Early Transcendentals. 11th ed. John Wiley & Sons.
[2] Edwards, C. H. (2019). Calculus. 10th ed. Pearson Education.
[3] Larson, R. (2020). Calculus. 11th ed. Cengage Learning.

Note: The references provided are for illustrative purposes only and are not intended to be a comprehensive list of resources on the topic.

Introduction

In our previous article, we explored how to eliminate the parameter to find a simplified Cartesian equation. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the purpose of eliminating the parameter?

A: The purpose of eliminating the parameter is to express the parametric equations in terms of a single variable, usually x or y, and then obtain a linear equation in the form of y = mx + b. This process is useful in various applications, such as graphing parametric equations, finding the equation of a tangent line, and solving systems of equations.

Q: How do I know if I can eliminate the parameter?

A: You can eliminate the parameter if the parametric equations are linear and the parameter is a single variable. In other words, if the parametric equations can be expressed in the form of x(t) = at + b and y(t) = ct + d, where a, b, c, and d are constants, then you can eliminate the parameter.

Q: What are the steps to eliminate the parameter?

A: The steps to eliminate the parameter are:

Solve one of the parametric equations for the parameter.
Substitute the expression for the parameter into the other parametric equation.
Simplify the resulting equation to obtain a linear equation in the form of y = mx + b.

Q: Can I eliminate the parameter if the parametric equations are non-linear?

A: No, you cannot eliminate the parameter if the parametric equations are non-linear. In this case, you will need to use other methods, such as substitution or elimination, to solve the system of equations.

Q: What are some common mistakes to avoid when eliminating the parameter?

A: Some common mistakes to avoid when eliminating the parameter are:

Not solving one of the parametric equations for the parameter.
Not substituting the expression for the parameter into the other parametric equation.
Not simplifying the resulting equation to obtain a linear equation in the form of y = mx + b.

Q: Can I use technology to eliminate the parameter?

A: Yes, you can use technology, such as graphing calculators or computer algebra systems, to eliminate the parameter. These tools can help you solve the system of equations and obtain a linear equation in the form of y = mx + b.

Q: What are some real-world applications of eliminating the parameter?

A: Some real-world applications of eliminating the parameter include:

Graphing parametric equations
Finding the equation of a tangent line
Solving systems of equations
Modeling real-world phenomena, such as the motion of an object

Q: Can I use eliminating the parameter to solve systems of equations with more than two variables?

A: No, you cannot use eliminating the parameter to solve systems of equations with more than two variables. In this case, you will need to use other methods, such as substitution or elimination, to solve the system of equations.

Conclusion

In conclusion, eliminating the parameter is a useful technique for expressing parametric equations in terms of a single variable and obtaining a linear equation in the form of y = mx + b. By following the steps outlined in this article, you can eliminate the parameter and solve a wide range of problems in mathematics and science.

References

[1] Anton, H. (2018). Calculus: Early Transcendentals. 11th ed. John Wiley & Sons.
[2] Edwards, C. H. (2019). Calculus. 10th ed. Pearson Education.
[3] Larson, R. (2020). Calculus. 11th ed. Cengage Learning.

Note: The references provided are for illustrative purposes only and are not intended to be a comprehensive list of resources on the topic.