Chapter 1 Linear Equations
1.3 Graph Linear Equations in Two Variables
Learning Objectives
By the end of this section, you will be able to:
- Plot points in a rectangular coordinate system
- Graph a linear equation by plotting points
- Graph vertical and horizontal lines
- Find the x– and y-intercepts
- Graph a line using the intercepts
Plot Points on a Rectangular Coordinate System
Just like maps use a grid system to identify locations, a grid system is used in mathematics to locate points in a plane. This grid system is called the rectangular coordinate system, or the Cartesian coordinate system, which is named after René Descartes, the French mathematician who is credited with developing this grid system in the seventeenth century.
The rectangular coordinate system is formed by two intersecting number lines, one horizontal and one vertical. The horizontal number line is called the x-axis. The vertical number line is called the y-axis. These axes divide a plane into four regions, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise.
In the rectangular coordinate system, every point is represented by an ordered pair. The first number in the ordered pair is the x-coordinate of the point, and the second number is the y-coordinate of the point. The phrase “ordered pair” means that the order is important.
Ordered Pair
An ordered pair [latex](x,y)[/latex] gives the coordinates of a point in a rectangular coordinate system. The first number is the x-coordinate. The second number is the y-coordinate.
What is the ordered pair of the point where the axes cross? At that point both coordinates are zero, so its ordered pair is [latex](0,0)[/latex]. The point [latex](0,0)[/latex] has a special name. It is called the origin.
The point [latex](0,0)[/latex] is called the origin. It is the point where the x-axis and y-axis intersect.
We use the coordinates to locate a point on the xy-plane. Let’s plot the point [latex](1,3)[/latex] as an example. First, locate 1 on the x-axis and lightly sketch a vertical line through [latex]x=1[/latex]. Then, locate 3 on the y-axis and sketch a horizontal line through [latex]y=3[/latex]. Now, find the point where these two lines meet—that is the point with coordinates [latex](1,3)[/latex].
Notice that the vertical line through [latex]x=1[/latex] and the horizontal line through [latex]y=3[/latex] are not part of the graph. We just used them to help us locate the point [latex](1,3)[/latex].
When one of the coordinates is zero, the point lies on one of the axes. In the figure below, the point [latex](0,4)[/latex] is on the y-axis and the point [latex](-2,0)[/latex] is on the x-axis.
Points on the Axes
Points with a y-coordinate equal to 0 are on the x-axis and have coordinates [latex](a,0)[/latex].
Points with an x-coordinate equal to 0 are on the y-axis and have coordinates [latex](0,b)[/latex].
Example 1
Plot each point in the rectangular coordinate system