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What is a Cartesian Equation? Definition, Formula & Examples

Home » What is a Cartesian Equation? Definition, Formula & Examples

A Cartesian equation is a mathematical equation that describes the relationship between variables in terms of their coordinates on a Cartesian plane. This plane is a two-dimensional coordinate system that consists of a horizontal x-axis and a vertical y-axis to locate points in space. In the Cartesian coordinate system, each point is identified by an ordered pair of coordinates, (x, y). Here, x represents the horizontal distance from the y-axis, and y represents the vertical distance from the x-axis. The origin (0, 0) is the point where the two axes (x and y) meet. A Cartesian equation usually involves polynomials in x and y. The solutions to the equation are the points on the Cartesian plane where the equation is true. Read this blog to learn more about ‘what is a Cartesian Equation?’ Here, we have shared the definition, history, and formula of Cartesian equations with examples.

What is the Origin of Cartesian Equations?

The term “Cartesian” was coined by Rene Descartes, a French philosopher and mathematician, in 1637. Meanwhile, Pierre de Fermat discovered it independently, but didn’t publish his findings. Another mathematician, Nicole Oresme, had also used similar concepts centuries before Fermat and Descartes.

The Cartesian system has grown and become a key part of calculus, developed by Gottfried Wilhelm Leibniz and Isaac Newton. It uses a simple grid with two coordinates (x and y) to describe points on a plane. This idea has been extended and used in more advanced mathematical concepts, like vector spaces.

Definition of a Cartesian Equation

A Cartesian equation is a simple equation that uses xs and ys variables to draw a curve on a graph. However, sometimes it will be hard to find the equation of a curve. That is where parametric equations come in.

Parametric equations use a third variable, usually “t”, to help find the equation of a curve instantly. For example, if we have y = 4t, we can divide both sides by 4 to get (1/4)y = t.

This will help us find the equation of the curve more easily. Typically, parametric equations are useful for drawing curves and surfaces on a graph using their Cartesian coordinates (x, y).

Examples of Cartesian Equations

For your better understanding, below we have presented two Cartesian Equation examples

Example 1: Converting Parametric Equations

Given parametric equations:

x = 2 + t^2
y = 4t

To convert to Cartesian equation:

  1. Solve for t: y = 4t → t = (1/4)y
  2. Substitute t into x equation: x = 2 + (1/4(y))^2
  3. Simplify: x = 2 + 1/16 y^2
  4. Rearrange to standard form: y = √(16x – 32)

Example 2: Equation of a Circle

The equation of a circle with center (2,3) and radius 4 is:

(x -2)^2 + (y-3)^2 = 16

This equation represents all points (x, y) on the circle.

  1. (x-2)^2 measures the horizontal distance between the point and the circle’s center..
  2. (y-3)^2 measures the vertical distance between the point and the circle’s center.
  3. The sum of these distances equals the radius squared (16)

The Importance of Cartesian Equations

Cartesian equations are a fundamental tool in mathematics. The Cartesian equations are widely used in areas such as algebra, geometry, calculus, and physics. Furthermore, they also help to describe and analyze relationships between variables. Therefore, it is considered to be a crucial concept in modern mathematics.

What are the Dimensions of a Cartesian Equation?

What is a Cartesian Equation

The Cartesian coordinate system is a fundamental concept in mathematics that allows us to represent points in space using numerical coordinates. This system is applicable in various dimensions, including one, two, and three dimensions.

One Dimension

In a one-dimensional space, the Cartesian coordinate system is established by:

  • Selecting a point O as the origin on a straight line
  • Choosing a unit of length
  • Defining an orientation for the line (i.e., determining which direction is positive and which is negative)

This one-dimensional Cartesian system is commonly referred to as the number line. Every real number has a corresponding location on this line, and each point on the line can be associated with a real number.

Two Dimension

In two-dimensional space, the Cartesian coordinate system consists of:

  • Two perpendicular axes (x-axis and y-axis)
  • A common origin point (O) where the axes intersect
  • A unit of length for both axes
  • An orientation for each axis

This two-dimensional Cartesian system enables us to represent points on a plane using ordered pairs of coordinates (x, y).

Three Dimension

In three-dimensional space, the Cartesian coordinate system comprises:

  • Three mutually perpendicular axes (x-axis, y-axis, and z-axis)
  • A common origin point (O) where the axes intersect
  • A unit of length for all three axes
  • An orientation for each axis

In this three-dimensional Cartesian system, points in space are represented using ordered triples of coordinates (x, y, z).

What is the Cartesian Equation Formula?

Cartesian equations are mathematical representations of relationships between variables in different dimensions. Here are the key formulas:

3D Space: Plane Equation

The Cartesian equation for a plane in three-dimensional space is:

Ax + By + Cz + D = 0

where

A, B, C, and D are constants

x, y, and z are variables representing points in 3D space

2D Space: Linear Equation

In two-dimensional space, the Cartesian equation for a straight line is:

y = mx + c

where:

m is the slope (gradient) of the line

c is the y-intercept (point where the line crosses the y-axis)

x and y are variables representing points on the line

Application of a Cartesian Equation

To use Cartesian coordinates in a practical scenario, consider the following three key steps:

  1. Define the Reference Point: Assign a specific location or landmark as the origin.
  2. Set the Scale: Determine the units of measurement to represent distances.
  3. Establish Orientation: Use available directional cues to define the orientation of the axes, leaving one pair’s orientation to be determined.

Final Words

By now, you will have learned what a Cartesian equation means. In summary, the Cartesian equation is a mathematical representation of geometric shapes like lines, circles, and curves using algebra. It uses a 2D coordinate system with x and y axes to pinpoint locations as (x, y) pairs. A Cartesian equation typically combines x and y in an algebraic expression, such as y = mx + b for lines, or x^2 + y^2 = r^2 for circles. By plotting points that satisfy these equations, you can visualize the corresponding geometric shapes.

If you are still not clear what a Cartesian equation is and are unsure how to solve mathematical problems related to it, call us. The Math assignment experts from our platform will guide you in applying the Cartesian equation formula and accurately solving problems based on it.

Jay Cooper Education Reading Time: 6 minutes

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