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The x-intercept is the point where the graph intersects the x-axis and the y-intercept is the point where the graph intersects the y-axis. These points have the form (x, 0) and (0, y) respectively.

To find the x- and y-intercepts algebraically, we use the fact that all x-intercepts have a y-value of zero and all y-intercepts have an x-value of zero. For example,

Graph: 3x − 5y = 15

Tip 1: To find the y-intercept, set x = 0 and determine the corresponding y-value. Similarly, to find the x-intercept we set y = 0 and determine the corresponding x-value.

Keep in mind that the intercepts are ordered pairs and not numbers. In other words, the x-intercept is not x = 5 but rather (5, 0).

Two points determine a line. If we find the x- and y-intercepts, then we can use them to graph the line. As you can see, they are fairly easy to find. Plot the points and draw a line through them with a straightedge.

Done. Let’s do another one.

Graph: y = −x + 9

We begin by finding the x-intercept.

The x-intercept is (3, 0).

The y-intercept is (0, 9). Now graph the two points.

Tip 2: Use Desmos.com to check your answer – it’s totally free. Just type in the equation.

This is a nice and easy method for determining the two points you need for graphing a line. In fact, we will use this exact technique for finding intercepts when we study the graphs of all the conic sections later in our study of Algebra.

Graph −4x + 3y = 12 using the intercepts.

Graph −4x + 2y = −6 using the intercepts.

Graph y = −5x +15 using the intercepts.

Graph y = −3/4 x + 9 using the intercepts.

This brings us to one of the most popular questions in linear graphing. Do all lines have x- and y-intercepts? The answer is NO. Horizontal lines, of the form y = b, do not necessarily have x-intercepts. Vertical lines, of the form x = a, do not necessarily have y-intercepts.

Graph y = 3.

Graph x = −2.

Many students this method, but I will tell you, there is a better way. Even less work... [ Graph Lines using Slope and Intercepts ] Read on!
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One way to graph lines is by plotting points. You just need to find the points to plot.

Graph: 3x − 5y = 15

We begin by solving for the variable y.

To do this, you need to do some algebra. Be careful here, this is where students often make errors.

After isolating the variable y, we now have an equation where y depends on x. That is, x is the independent variable and y is the dependent variable.

Tip 1: When solving, be sure to divide BOTH terms on the right side by the coefficient of y. In the example above, we divided −5 into the terms −3x and 15 separately.

Now we have an equivalent equation, which will be easier to work with. Get used to fractions, many equations of lines have fractions in them.

Graph: y = 3/5 x − 3

When graphing by plotting points, teachers typically require that you to plot at least five points. To find these points, you will choose any x-values and then substitute them into the equation to find the corresponding y-values.

Tip 2: Choose some negative x-values, zero, and some positive x-values.

Tip 3: Avoid fractions by choosing x-values wisely.

Since the denominator of 3/5 is 5, we will choose multiples of 5 to avoid fractions. In this case, we choose −5, 0, 5, 10, and 15 for the x-values. Make a table,

Then substitute these x-values into the equation to find the corresponding y-values.

The table gives us 5 ordered pairs to plot. Since x-values here are multiples of 5, we choose the tick on the x-axis to represent 5 units. Similarly because the y-values are multiples of 3, we will choose a scale of 3 units on the y-axis.

Tip 4: Impress your teacher by placing an arrow on either end of the line to indicate that it continues forever.

That’s it! The general steps are outlined below.

Step 1: Solve for y so that your equation looks like y = mx + b.
Step 2: Choose any five x-values. (You only really need two.)
Step 3: Plug in to find the corresponding y-values.
Step 4: Plot the points and connect them with a straightedge.

Here is another example.

Graph: −2x − 3y = 6

First solve for y.

Tip 5: Avoid the following common error of dividing only one term.

When dividing a binomial by a number you must divide both terms by that number. Next, choose some x-values. Avoid fractions here by choosing −6, −3, 0, 3, and 6. Substituting we have,

And plotting these points we have the graph of y = −2/3 x − 2,

Some more examples follow.

Graph: y = 2x − 4 by plotting five points.

Graph: y = −x − 2 by plotting five points.

Choosing a scale when creating a blank coordinate system will take some thought. Keep in mind that the scale on the x-axis need not be the same as the scale on the y-axis.

Graph: y = 1/2 x − 6 by plotting five points.

When the coefficient of x is a fraction, choose x-values to be multiples of the denominator so that you might avoid unnecessarily tedious calculations.

Graph: y = −3/2 x + 6 by plotting five points.