How to Graph Linear Equations



This Instructable will teach you the three main ways to graph linear equations - the table method, the intercept method, and the slope-intercept formula. You can either watch the Youtube video or read the instructions below.

If you want to see more math tutorials, you can check out my YouTube account, Math for Noobs!

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Step 1: Defining Terms

Before we begin, we should define some terms:

  1. Equation of a line: the equation of a line is typically written as y=mx+b (this is called the slope-intercept form where m is the slope and b is the y intercept). Other forms include the standard form (ax+by=c, where a is a positive integer and b and c are integers) and the point-slope form ( y-y1=m(x-x1), where m is the slope and (x1,y1) is a point on the line). If you want more info on this, you can watch my video here.
  2. Slope: the slope shows how steep the line is. It is defined as the change in y over the change in x, or rise over run. The steeper the line, the greater the value of the slope. A positive slope goes up from left to right, and a negative slope goes down from left to right. For more info on slope, watch my video here.
  3. Intercept: The x-intercept is where the line crosses the x-axis, and the y-intercept is where the line crosses the y-axis. For more info on intercepts, watch my video here.

Step 2: The Table Method

For the table method, you must create an x/y table, find points on the graph, plot the points, and connect the dots. This method works for any type of equation (not only lines), but it is the most time consuming.

  1. Draw an x/y table.
  2. Next we must find a few points on the line. In order to do this, first choose a few numbers and write them in the x column. Typically, you don't want to choose the numbers completely randomly - you want to choose a few target points - for a parabola, for example, you want to choose numbers near the vertex of the parabola. But since this is a line, any point will do. But note that choosing small numbers will make the calculations easier. Also, you want to choose numbers within the boundaries of the coordinate plane you are given. For this problem, I chose -1, 0, 1, 2, and 3.
  3. Plug each x value into the equation, and write the y value in the y column. My equation was y=x-2, and the y values I got were -3, -2, -1, 0, and 1. For example, for x = -1, we plug in -1 to the equation. y=-1-2 = -3. Therefore, we write -3 next to -1.
  4. Plot the points on the graph. The coordinates of the points are (x value, y value). So the points I plotted were (-1,-3), (0,-2), (1, -1), (2, 0), and (3, 1).
  5. Connect the dots on the graph to create the line.

Step 3: The Intercept Method

For the intercept method, you only need to find 2 points - the x-intercept and the y-intercept. This method works best when the intercepts are whole numbers. Although this method may not produce a very accurate line (when the intercepts are fractions, you must estimate the exact location of the points) it is very quick and efficient.

  1. Find the x intercept. In order to find the x-intercept, plug y=0 into the equation and calculate to find x. You must plug in y=0 because the x-intercept is where the line crosses the x axis, and the y value is always zero on the x axis. The coordinates of the x-intercept will then be (x value, 0).
    1. For the problem above, we plug in y=0 and get 0=x-2. We rearrange to get x=2. The x-intercept is therefore (2,0).
  2. Find the y intercept. This time plug x=0 into the equation and calculate to find y. We must plug in x=0 because x=0 at any point on the y axis. The coordinate of the y intercept is (0, y value).
    1. Plug in x=0 to the practice problem and we get y = 0 - 2 = -2. The y-intercept is (0,-2).
  3. Plot the two points.
  4. Connect the points.

Step 4: Slope - Intercept Formula Method

For the slope-intercept method, you need to first rearrange the equation into the form y=mx+b. Then you can plot b, the y-intercept, and find other points using m, the slope. In my opinion, this method is the most efficient, especially if the equation is already written in the slope-intercept form.

  1. Rearrange the equation into the slope-intercept form, y=mx+b. In the problem above, y=x-2 was already in slope-intercept form so we do not have to do anything for this step.
  2. Find b, the y-intercept, and plot the point on the graph. In this case, the y-intercept is -2 so we must plot the point (0,-2).
  3. Find m, or the slope. Since the slope is the change in y over the change in x (or rise over run), we can plot the next point by using the slope. In the example problem, the slope is 1, which equals 1/1. So the change in y over the change in x is 1 over 1. This means that from the y-intercept (0,-2), we must move one unit up and one unit to the right to find another point on the graph. This brings us to (1,-1). We keep repeating by moving one unit up and one unit to the right. Once we have enough points, we can connect the dots and create the line.
  4. Note: Moving 1 unit up, 1 unit to the right is not the only option. In the practice problem, the slope is 1, which also equals -1/-1, 2/2, 3/3, etc. So we can use any of these fractions to graph the line. For example, if we choose to use -1/-1, we can move one unit down and one unit to the left from the y-intercept. Either way, we would still get the same line.

Step 5: More Practice!

Try graphing these:

  • y = (1/5)x - 4
  • x + y = -3
  • 6x - 3y = 1

If you want to check your answers, either graph the equations on a calculator and compare the result to your answer, or watch my video with the solutions here. The problems begin at 5:24.

Thanks for reading and if you have any questions, feel free to comment!

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