Cartesian Coordinate System

Definition

The Cartesian coordinate system is a two-dimensional coordinate system that uses perpendicular lines, called axes, to specify the location of points in a plane. It is named after the French mathematician and philosopher René Descartes, who introduced this system as a way to represent and analyze geometric relationships algebraically.

5 Must Know Facts For Your Next Test

1. The Cartesian coordinate system is used to represent and analyze the relationships between variables in a variety of mathematical and scientific contexts, including graphing linear and nonlinear functions.
2. The x-axis is the horizontal axis, and the y-axis is the vertical axis, with the origin (0, 0) representing the point where the two axes intersect.
3. The coordinates of a point on the coordinate plane are expressed as an ordered pair (x, y), where x represents the horizontal position and y represents the vertical position.
4. The four quadrants of the coordinate plane are defined by the signs of the x and y coordinates: Quadrant I (+, +), Quadrant II (-, +), Quadrant III (-, -), and Quadrant IV (+, -).
5. The Cartesian coordinate system is essential for solving systems of linear equations by graphing, as the intersection of the lines represents the solution to the system.

Review Questions

• Explain how the Cartesian coordinate system is used to represent and analyze the relationships between variables in mathematical and scientific contexts.
  • The Cartesian coordinate system provides a way to represent and analyze the relationships between variables by using a two-dimensional plane with perpendicular axes, the x-axis and the y-axis. This system allows for the precise location of points on the plane, which can be used to graph functions, plot data, and visualize the relationships between different variables. For example, in the context of solving systems of linear equations by graphing, the Cartesian coordinate system is essential for representing the equations as lines and finding the point of intersection, which represents the solution to the system.
• Describe the key features of the Cartesian coordinate system, including the x-axis, y-axis, origin, and quadrants.
  • The Cartesian coordinate system is defined by two perpendicular axes: the horizontal x-axis and the vertical y-axis. The point where these two axes intersect is called the origin, with coordinates (0, 0). The coordinate plane is divided into four quadrants, numbered I, II, III, and IV, based on the signs of the x and y coordinates in each region. Quadrant I has positive x and y coordinates, Quadrant II has negative x and positive y, Quadrant III has negative x and y, and Quadrant IV has positive x and negative y. The location of a point on the coordinate plane is specified by an ordered pair (x, y), where x represents the horizontal position and y represents the vertical position.
• Analyze how the Cartesian coordinate system is essential for solving systems of linear equations by graphing, and explain the significance of the point of intersection.
  • The Cartesian coordinate system is fundamental for solving systems of linear equations by graphing because it provides a visual representation of the equations as lines on the coordinate plane. Each linear equation in the system can be graphed as a line, and the point where these lines intersect represents the solution to the system. The coordinates of this point of intersection, given as an ordered pair (x, y), satisfy all the equations in the system simultaneously, making it the unique solution. The ability to graph and visualize the relationships between the equations in a system is a key advantage of using the Cartesian coordinate system to solve systems of linear equations, as it allows for a deeper understanding of the problem and the identification of the solution.

Related terms

Coordinate Plane: The coordinate plane is the two-dimensional surface defined by the Cartesian coordinate system, with the horizontal axis (x-axis) and the vertical axis (y-axis) intersecting at the origin (0, 0).

Quadrants: The coordinate plane is divided into four quadrants, numbered I, II, III, and IV, based on the signs of the x and y coordinates in each region.

Ordered Pair: An ordered pair is a set of two numbers that represents the location of a point on the coordinate plane, with the first number (x-coordinate) representing the horizontal position and the second number (y-coordinate) representing the vertical position.