Introduction to Coordinate Geometry

Coordinate geometry is a very important area of math. It combines algebra with geometry. It is based on idea that algebra equations can be represented as geometric shapes. This blog also has algebra and geometry as two separate categories.

There is an amazing regularity between degree of equations and the geometric shapes they form. All linear equations in two variables give straight lines. Lines formed by linear equations differ only in their positioning in a coordinate plane. This article includes example problems involving only simple linear equations.

Similarly all quadratic equations in two variables give simple curves. Higher degree equations also behave in a similar way.

Coordinate Plane

The plane uses two lines perpendicular to each other for identifying points. Here is the diagrammatic representation of a coordinate plane.



As shown in the figure, a point is identified based on where it is positioned with reference to both horizontal x axis and vertical y axis. The two axes are like  two number lines that share a common origin O. Each point in the plane is represented as a unique pair of numbers. The horizontal distance from origin is x coordinate and is mentioned first. The vertical distance from origin is y coordinate and is mentioned second. For the origin O both these distance are zero. Hence its coordinates are (0,0).

Consider the point shown as red dot and labeled (3,2). It is easy to see that its horizontal distance from origin is 3 and vertical distance from origin is 2.  The two red dashes lines show this clearly.

As on a number line, coordinates are negative on the other side of 0. The point shown in blue has a negative y coordinate.

Example 1

Graph the linear equation 2x + y = 3.

2x + y = 3

y = 3 2x

When x = 0

y = 3 2x

y = 3 2 × 0

y = 3 0

y = 3

Point  (0, 3) satisfies the equation and is one of its solutions.

Similarly when x = –1 then  y = 5 and when  x = 1 then  y = 1. These values for y can be determined from the equation. This has been done here for x = 0. Try to determine these two y values by yourself using the equation.

So points (–1, 5) and (1, 1) are also solutions to the equation. The three points are marked as red dots and joined in the graph below. A straight line is formed.


Example 2

Use the graph in example 1 to find y when x = 0.5.


A vertical line from the point x = 0.5 to the graph line identifies the point on the graph where x = 0.5. Another horizontal line from this point on the graph to y axis identifies the y coordinate of the point which is 2. So when  x = 0.5 then y = 2. The coordinates of the blue point are (0.5 , 2).

For any known x or y value, the matching value of the other variable can be found from the graph in a similar way. Dashes lines are used to differentiate these lines from the main graph line that is solid.

The amazing relationship between equations and their graphic representation carries math to a higher level. It is important to consider this connection very closely. Thinking more about this link can help your math learning.

Here is a link to an excellent introduction to coordinate geometry on MathOpenRef. The site has more articles on coordinate geometry beside this introduction.

Click Here to Read Introduction to Coordinate Geometry on MathOpenRef

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