An old story describes how seventeenth-century philosopher/mathematician René Descartes invented the system that has become the foundation of algebra while sick in bed. According to the story, Descartes was staring at a fly crawling on the ceiling when he realized that he could describe the fly’s location in relation to the perpendicular lines formed by the adjacent walls of his room. He viewed the perpendicular lines as horizontal and vertical axes. Further, by dividing each axis into equal unit lengths, Descartes saw that it was possible to locate any object in a two-dimensional plane using just two numbers—the displacement from the horizontal axis and the displacement from the vertical axis.

While there is evidence that ideas similar to Descartes’ grid system existed centuries earlier, it was Descartes who introduced the components that comprise the Cartesian coordinate system, a grid system having perpendicular axes. Descartes named the horizontal axis the x-axis and the vertical axis the y-axis.

The Cartesian coordinate system, also called the rectangular coordinate system, is based on a two-dimensional plane consisting of the x-axis and the y-axis. Perpendicular to each other, the axes divide the plane into four sections. Each section is called a quadrant; the quadrants are numbered counterclockwise as shown in the figure below.

The Cartesian coordinate system with all four quadrants labeled.

The center of the plane is the point at which the two axes cross. It is known as the origin or point [latex]\left(0,0\right)[/latex]. From the origin, each axis is further divided into equal units: increasing, positive numbers to the right on the x-axis and up the y-axis; decreasing, negative numbers to the left on the x-axis and down the y-axis. The axes extend to positive and negative infinity as shown by the arrowheads in the figure below.

Each point in the plane is identified by its x-coordinate, or horizontal displacement from the origin, and its y-coordinate, or vertical displacement from the origin. Together we write them as an ordered pair indicating the combined distance from the origin in the form [latex]\left(x,y\right)[/latex]. An ordered pair is also known as a coordinate pair because it consists of x and y-coordinates. For example, we can represent the point [latex]\left(3,-1\right)[/latex] in the plane by moving three units to the right of the origin in the horizontal direction and one unit down in the vertical direction.

An illustration of how to plot the point (3,-1).

When dividing the axes into equally spaced increments, note that the x-axis may be considered separately from the y-axis. In other words, while the x-axis may be divided and labeled according to consecutive integers, the y-axis may be divided and labeled by increments of 2 or 10 or 100. In fact, the axes may represent other units such as years against the balance in a savings account or quantity against cost. Consider the rectangular coordinate system primarily as a method for showing the relationship between two quantities.

Graphing Linear Equations by Using a Table

We can plot a set of points to represent an equation. When such an equation contains both an x variable and a y variable, it is called an equation in two variables. Its graph is called a graph in two variables. Any graph on a two-dimensional plane is a graph in two variables.

Suppose we want to graph the equation [latex]y=2x - 1[/latex]. We can begin by substituting a value for x into the equation and determining the resulting value of y. Each pair of x and y-values is an ordered pair that can be plotted. The table below lists values of x from –3 to 3 and the resulting values for y.

We can plot these points from the table. The points for this particular equation form a line, so we can connect them. This is not true for all equations.

Note that the x-values chosen are arbitrary regardless of the type of equation we are graphing. Of course, some situations may require particular values of x to be plotted in order to see a particular result. Otherwise, it is logical to choose values that can be calculated easily, and it is always a good idea to choose values that are both negative and positive. There is no rule dictating how many points to plot, although we need at least two to graph a line. Keep in mind, however, that the more points we plot, the more accurately we can sketch the graph.

Try It

Construct a table and graph the equation by plotting points: [latex]y=\frac{1}{2}x+2[/latex].

Show Solution

[latex]x[/latex]	[latex]y=\frac{1}{2}x+2[/latex]	[latex]\left(x,y\right)[/latex]
[latex]-2[/latex]	[latex]y=\frac{1}{2}\left(-2\right)+2=1[/latex]	[latex]\left(-2,1\right)[/latex]
[latex]-1[/latex]	[latex]y=\frac{1}{2}\left(-1\right)+2=\frac{3}{2}[/latex]	[latex]\left(-1,\frac{3}{2}\right)[/latex]
[latex]0[/latex]	[latex]y=\frac{1}{2}\left(0\right)+2=2[/latex]	[latex]\left(0,2\right)[/latex]
[latex]1[/latex]	[latex]y=\frac{1}{2}\left(1\right)+2=\frac{5}{2}[/latex]	[latex]\left(1,\frac{5}{2}\right)[/latex]
[latex]2[/latex]	[latex]y=\frac{1}{2}\left(2\right)+2=3[/latex]	[latex]\left(2,3\right)[/latex]

Using Intercepts to Plot Lines in the Coordinate Plane

The intercepts of a graph are points where the graph crosses the axes. The x-intercept is the point where the graph crosses the x-axis. At this point, the y-coordinate is zero. The y-intercept is the point where the graph crosses the y-axis. At this point, the x-coordinate is zero.

To determine the x-intercept, we set y equal to zero and solve for x. Similarly, to determine the y-intercept, we set x equal to zero and solve for y. For example, lets find the intercepts of the equation [latex]y=3x - 1[/latex].

To find the x-intercept, set [latex]y=0[/latex].

To find the y-intercept, set [latex]x=0[/latex].

We can confirm that our results make sense by observing a graph of the equation. Notice that the graph crosses the axes where we predicted it would.

Example: Finding the Intercepts of the Given Equation

Find the intercepts of the equation [latex]y=-3x - 4[/latex]. Then sketch the graph using only the intercepts.

Show Solution

Set [latex]y=0[/latex] to find the x-intercept.

[latex]\begin{array}{l}y=-3x - 4\hfill \\ 0=-3x - 4\hfill \\ 4=-3x\hfill \\ -\frac{4}{3}=x\hfill \\ \left(-\frac{4}{3},0\right)x\text{-intercept}\hfill \end{array}[/latex]

Set [latex]x=0[/latex] to find the y-intercept.

[latex]\begin{array}{l}y=-3x - 4\hfill \\ y=-3\left(0\right)-4\hfill \\ y=-4\hfill \\ \left(0,-4\right)y\text{-intercept}\hfill \end{array}[/latex]

Plot both points and draw a line passing through them.

Try It

Find the intercepts of the equation and sketch the graph: [latex]y=-\frac{3}{4}x+3[/latex].

Show Solution

x-intercept is [latex]\left(4,0\right)[/latex]; y-intercept is [latex]\left(0,3\right)[/latex].

Using a Graphing Utility to Plot Lines

https://www.desmos.com/calculator

https://www.mathway.com/Graph

https://www.symbolab.com/graphing-calculator

The Slope of a Line

The slope of a line refers to the ratio of the vertical change in y over the horizontal change in x between any two points on a line. It indicates the direction in which a line slants as well as its steepness. Slope is sometimes described as rise over run.

If the slope is positive, the line slants upward to the right. If the slope is negative, the line slants downward to the right. As the slope increases, the line becomes steeper. Some examples are shown below. The lines indicate the following slopes: [latex]m=-3[/latex], [latex]m=2[/latex], and [latex]m=\frac{1}{3}[/latex].