Cartesian Equation — Definition, Formula & Examples

A Cartesian equation is an equation that describes a curve or shape using the standard variables

x

and

y

(and sometimes

z

) in a coordinate system. For example,

y = 2x + 3

and

x^2 + y^2 = 25

are both Cartesian equations.

A Cartesian equation is an implicit or explicit algebraic relation $f(x, y) = 0$ (or $f(x, y, z) = 0$ in three dimensions) that defines a locus of points in a Cartesian coordinate system, without the use of a parameter.

How It Works

You write a Cartesian equation by expressing the relationship between

x

and

y

directly. If you start with parametric equations like

x = g(t)

and

y = h(t)

, you eliminate the parameter

t

to get one equation in

x

and

y

alone. The result is the Cartesian equation of the same curve. Any point

(x, y)

that satisfies the equation lies on the curve, and any point that does not satisfy it lies off the curve.

Worked Example

Problem: A curve is defined parametrically by

x = 3t

and

y = t^2

. Find its Cartesian equation.

Step 1: Solve the first equation for the parameter

t

t = \frac{x}{3}

Step 2: Substitute this expression into the second equation to eliminate

t

y = \left(\frac{x}{3}\right)^2 = \frac{x^2}{9}

Answer: The Cartesian equation of the curve is

y = \dfrac{x^2}{9}

, which is a parabola.

Why It Matters

Converting to Cartesian form lets you identify the type of curve (line, circle, parabola, etc.) at a glance. This skill appears constantly in precalculus, calculus, and physics whenever you need to graph or analyze motion described by parametric or polar equations.

Common Mistakes

Mistake: Confusing Cartesian equations with parametric equations and leaving the parameter

t

in the final answer.

Correction: A Cartesian equation must contain only

x

and

y

(no parameter). Always eliminate

t

completely before stating your answer.