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 belowlists values of *x* from –3 to 3 and the resulting values for *y*.

[latex]x[/latex] | [latex]y=2x - 1[/latex] | [latex]\left(x,y\right)[/latex] |

[latex]-3[/latex] | [latex]y=2\left(-3\right)-1=-7[/latex] | [latex]\left(-3,-7\right)[/latex] |

[latex]-2[/latex] | [latex]y=2\left(-2\right)-1=-5[/latex] | [latex]\left(-2,-5\right)[/latex] |

[latex]-1[/latex] | [latex]y=2\left(-1\right)-1=-3[/latex] | [latex]\left(-1,-3\right)[/latex] |

[latex]0[/latex] | [latex]y=2\left(0\right)-1=-1[/latex] | [latex]\left(0,-1\right)[/latex] |

[latex]1[/latex] | [latex]y=2\left(1\right)-1=1[/latex] | [latex]\left(1,1\right)[/latex] |

[latex]2[/latex] | [latex]y=2\left(2\right)-1=3[/latex] | [latex]\left(2,3\right)[/latex] |

[latex]3[/latex] | [latex]y=2\left(3\right)-1=5[/latex] | [latex]\left(3,5\right)[/latex] |

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.

### How To: Given an equation, graph by plotting points

- Make a table with one column labeled
*x*, a second column labeled with the equation, and a third column listing the resulting ordered pairs. - Enter
*x-*values down the first column using positive and negative values. Selecting the*x-*values in numerical order will make graphing easier. - Select
*x-*values that will yield*y-*values with little effort, preferably ones that can be calculated mentally. - Plot the ordered pairs.
- Connect the points if they form a line.

### Example: Graphing an Equation in Two Variables by Plotting Points

Graph the equation [latex]y=-x+2[/latex] by plotting points.

### Try It

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

Show Show Solution

## 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

**is the point where the graph crosses the**

*y-*intercept*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].

[latex]\begin{array}{llllll}y=3x - 1\hfill & \hfill \\ 0=3x - 1\hfill & \hfill \\ 1=3x\hfill & \hfill \\ \frac{1}{3}=x\hfill & \hfill \\ \left(\frac{1}{3},0\right)\hfill & x\text{-intercept}\hfill \end{array}[/latex]

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

[latex]\begin{array}{lllll}y=3x - 1\hfill & \hfill \\ y=3\left(0\right)-1\hfill & \hfill \\ y=-1\hfill & \hfill \\ \left(0,-1\right)\hfill & y\text{-intercept}\hfill \end{array}[/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.

### How To: Given an equation, find the intercepts

- Find the
*x*-intercept by setting [latex]y=0[/latex] and solving for [latex]x[/latex]. - Find the
*y-*intercept by setting [latex]x=0[/latex] and solving for [latex]y[/latex].

### 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

### Try It

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

Show Solution

## Using a Graphing Utility to Plot Lines

You can use an online graphing tool to quickly plot lines. Watch this short video Tutorial to learn how.

### Try It

Desmos has a helpful feature that allows you to turn a constant (number) into a variable. Follow these steps to learn how:

- Graph the line [latex]y=-\frac{2}{3}x-\frac{4}{3}[/latex].
- On the next line enter[latex]y=-a x-\frac{4}{3}[/latex]. You will see a button pop up that says “add slider: a”, click on the button. You will see the next line populated with the variable a and the interval on which a can take values.
- What part of a line does the variable a represent? The slope or the y-intercept?

Here is ashort tutorial with more information about sliders.

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