Question

Eliminate the parameter to express the following parametric equations as a single equation in \(x\) and \(y\) $$x=\sqrt{t+1}, y=\frac{1}{t+1}$$

Short answer

Question: Eliminate the parameter \(t\) from the parametric equations \(x = \sqrt{t + 1}\) and \(y = \frac{1}{t + 1}\) to obtain a single equation in terms of \(x\) and \(y\). Answer: \(y = \frac{1}{x^2}\)

Step-by-step solution

Express t in terms of x

We start by expressing \(t\) in terms of \(x\) using the first equation: $$x = \sqrt{t + 1}$$ First, we will square both sides, we get: $$x^2 = t + 1$$ Now, we can express \(t\) in terms of \(x\) by isolating \(t\): $$t = x^2 - 1$$

Replace t with the expression in terms of x in the second equation

Next, we will substitute the expression we found for \(t\) in the second equation: $$y = \frac{1}{t + 1}$$ Replacing \(t\) with the expression in terms of \(x\): $$y = \frac{1}{(x^2 - 1) + 1}$$

Simplify and obtain the final equation in x and y

Finally, we will simplify the equation and write it as a single equation in \(x\) and \(y\): $$y = \frac{1}{x^2}$$ This is our final equation, expressing the relationship between \(x\) and \(y\) without any parameter.

Key concepts

Eliminating the Parameter

In the given parametric equations, we have two separate equations that involve a parameter, denoted as \( t \). Our goal is to write them as one equation in terms of \( x \) and \( y \), without using the parameter. This process is known as eliminating the parameter.
To eliminate the parameter, we first need to express the parameter \( t \) in terms of one of the other variables, typically \( x \) or \( y \). This involves solving one of the given parametric equations for \( t \), allowing us to substitute in the other equation. By doing this, we remove \( t \) entirely, leaving a direct relationship between \( x \) and \( y \).
This method is useful because it converts complex parametric forms into simpler standard forms that are easier to analyze.

Expressing Variables

Expressing variables is a crucial part of working with parametric equations. It allows you to describe one variable completely in terms of another. In our exercise, the first step involves solving for the parameter \( t \) using the equation \( x = \sqrt{t+1} \).
To express \( t \) in terms of \( x \):

• First, square both sides of the equation: \( x^2 = t + 1 \).
• Next, isolate \( t \) by subtracting 1 from both sides, resulting in \( t = x^2 - 1 \).

This expression of \( t \) in terms of \( x \) is key to eliminating the parameter. It provides us with a substitution that can be used in the second equation, allowing us to transition to a relationship solely between \( x \) and \( y \).
Recognizing how to switch between different forms by expressing variables in terms of each other is an essential algebraic skill.

Equation Manipulation

Equation manipulation involves altering equations to achieve a desired form, often by substituting expressions. This process is at the heart of eliminating parameters.
Once we have expressed \( t = x^2 - 1 \), the next step in our exercise is to substitute this expression for \( t \) into the second given equation \( y = \frac{1}{t+1} \). By replacing \( t \) in the equation, it becomes:

• \( y = \frac{1}{(x^2 - 1) + 1} \)

This substitution removes the parameter \( t \) from the equation, transforming it into a form that directly relates \( x \) to \( y \).
Equation manipulation is a broad concept encompassing various techniques used to simplify or solve equations. It is fundamental in both algebra and calculus, allowing for clarity and solvability.

Algebraic Simplification

After parameter elimination and equation manipulation, the final step is algebraic simplification. This enables us to write the cleanest form of the equation relating \( x \) and \( y \). The goal is a simple, direct expression that reveals the relationship clearly.
Continuing with our process, we took the transformed equation:

• \( y = \frac{1}{x^2 - 1 + 1} \)

We simplify the expression inside the fraction: \( x^2 - 1 + 1 = x^2 \). Therefore, the simplified equation is:

• \( y = \frac{1}{x^2} \)

This final expression of \( y \) as a function of \( x \) is simpler and devoid of the parameter \( t \).
Algebraic simplification is a key final step in any mathematical problem-solving process. It ensures that the equation is as understandable and useful as possible, allowing for easier interpretation and application of the results.