Parametric equations are equations in which *y* is a function of *x*, but both *x* and *y* are defined in terms of a third variable. The third variable is the parameter of the equations. Often, the variable *t* is used in this type of equation.

Here, we will learn about parametric equations with solved exercises. Also, we will look at some practice problems.

## Summary of parametric equations

In parametric equations, $latex y$ is defined as a function of $latex x$ by expressing both $latex y$ and $latex x$ in terms of a third variable known as a parameter.

For example, the following equations are parametric equations in which the parameter is $latex t$.

$latex x=t+1~~~[1]$

$latex y=t^2~~~[2]$

These parametric equations define the parabola with the equation:

$latex y=x^2-2x+1$

We can obtain this by eliminating the parameter $latex t$ in equations [1] and [2]. If we solve equation [1] for $latex t$, we have $latex t=x-1$. Substituting this into equation [2], we have:

$latex y=(x-1)^2$

$latex y=x^2-2x+1$

## Parametric equations – Examples with answers

**EXAMPLE **1

**EXAMPLE**

What is the Cartesian equation of the following parametric equations?

$latex x=\sqrt{t}$

$latex y=3t^2-4$

##### Solution

A Cartesian equation is an equation of *y* expressed in terms of *x*. Therefore, we have to eliminate the parameter $latex t$.

We start by solving the first equation for $latex t$. For this, we square both sides:

$latex x=\sqrt{t}$

$latex x^2=t$

Substituting this expression into the second equation, we have:

$latex y=3t^2-4$

$latex y=3(x^2)^2-4$

$latex y=3x^4-4$

**EXAMPLE **2

**EXAMPLE**Find the Cartesian equation for the following parametric equations:

$latex x=2t-1$

$latex y=12t^2-5$

##### Solution

To get a Cartesian equation, we have to get an equation of *y* in terms of *x*.

Then, we start by solving the first equation for $latex t$:

$latex x=2t-1$

$$t=\frac{x+1}{2}$$

Now, we substitute this expression into the second equation:

$latex y=12t^2-5$

$$y=12\left(\frac{x+1}{2}\right)^2-5$$

$$y=12\left(\frac{x^2+2x+1}{4}\right)-5$$

$$y=3x^2+6x+3-5$$

$latex y=3x^2+6x-2$

**EXAMPLE **3

**EXAMPLE**Express *y* as an equation of *x* using the following parametric equations:

$latex x=2\sqrt{t}$

$latex y=8t^2+5$

##### Solution

To express $latex y$ as an equation of $latex x$, we have to start by solving the first equation for $latex t$:

$latex x=2\sqrt{t}$

$$\sqrt{t}=\frac{x}{2}$$

$latex t=\frac{x^2}{4}$

Now, we use this expression in the second equation:

$latex y=8t^2+5$

$$y=8\left(\frac{x^2}{4}\right)^2+5$$

$$y=8\left(\frac{x^4}{16}\right)+5$$

$$y=\frac{x^4}{2}+5$$

**EXAMPLE **4

**EXAMPLE**Find the Cartesian equation for the following parametric equations:

$$x=\frac{1}{t}$$

$latex y=3t-2$

##### Solution

Solving the first equation for $latex t$, we have:

$$x=\frac{1}{t}$$

$$t=\frac{1}{x}$$

Substituting this expression in the second equation, we have:

$latex y=3t-2$

$$y=3\left(\frac{1}{x}\right)-2$$

$$y=\frac{3}{x}-2$$

We can multiply the whole equation by $latex x$ to get $latex xy+2x=3$.

**EXAMPLE **5

**EXAMPLE**Find an equation for *y* in terms of *x* using the equations below:

$$x=\frac{2}{\sqrt{x}}$$

$$y=\frac{3}{1+3}$$

##### Solution

We start by finding an expression for $latex t$ in terms of $latex x$:

$$x=\frac{2}{\sqrt{t}}$$

$$x^2=\frac{4}{t}$$

$$t=\frac{4}{x^2}$$

Now, we use this expression in the second equation:

$$y=\frac{3}{1+t}$$

$$y=\frac{3}{1+\frac{4}{x^2}}$$

To simplify, we can multiply both the numerator and the denominator by $latex x^2$:

$$y=\frac{3x^2}{x^2+4}$$

**EXAMPLE **6

**EXAMPLE**Find a Cartesian equation using the following parametric equations:

$$x=\frac{1}{2-t}$$

$$y=\frac{3}{1+2t}$$

##### Solution

The $latex x$ equation can be solved for $latex t$ as follows:

$$x=\frac{1}{2-t}$$

$latex x(x-t)=1$

$latex 2x-xt=1$

$latex xt=2x-1$

$$t=\frac{2x-1}{x}$$

Substituting this expression into the $latex y$ equation, we have:

$$y=\frac{3}{1+2t}$$

$$y=\frac{3}{1+2\left(\frac{2x-1}{x}\right)}$$

$$y=\frac{3x}{x+2(2x-1)}$$

$$y=\frac{3x}{5x-2}$$

## Parametric equations – Practice problems

#### The Cartesian equation of the equations $latex x=\frac{t}{1-3t}$, $latex y=\frac{t}{1+2t}$ is written as a fraction. What is the denominator?

Write the denominator in the input box.

## See also

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