Rearranging Linear Equations Before Graphing

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we will review slope and what it represents. We will then talk about the slope-intercept form of a line and how to use this form to graph a line. Lastly, we will look at how to rearrange equations to make graphing them very easy.

Slope

Suppose you are growing your hair out, and you want to know how quickly your hair is growing each month. You measure your hair for the first four months. After 2 months, your hair length is 6.5 inches, and after 4 months, your hair length is 8 inches.

The rate at which your hair is growing would be the slope of the line representing your hair growth. The slope of a line is the rate at which y is changing with respect to x. We can find the slope using the following formula:


rearrange1


In our hair example, we have that at 2 months, your hair is 6.5 inches, and at 4 months, your hair is 8 inches. This gives the two points, (2, 6.5) and (4, 8), on the line representing your hair growth. We can find the slope, or the rate at which your hair is growing, using the formula.


rearrange2


We see that your hair is growing at a rate of 3/4 inches per month, so the slope of the line representing your hair growth is 3/4.

What's great about knowing the slope of a line is that if we know a point on the line, we can use the slope to find another point on the line. If a line has slope a/b, we start at the known point and move a units vertically and b units horizontally to get to another point on the line. That is, if the point (x 1, y 1) is on a line with slope a/b, then the point (x 1 + b, y 1 + a) is also on the line. If we do this on a graph and then connect the dots in a line, we have a graph of the line!

Consider our hair example. We know two points on the line, so we can graph the line by connecting these points. The following image illustrates that if we start at the point (2, 6.5), and move up 3 units and over 4 units like the slope, 3/4, suggests, we end up with another point on that line.


rearrange8


By looking at our newly-graphed line, we can see at the six month mark, your hair length will be 9.5 inches. We see that if we know the slope and a point on a line, we can graph the line!

Slope-Intercept Form of a Line

Okay, great, so we know how to use a point on a line and the line's slope to graph the line, but how does that help us to graph a line given to us in equation form? If we could somehow identify the slope and a point on the line from an equation, we could use what we just learned to graph the line.

I've got great news! There is a form of a linear equation called slope-intercept form, and we can easily identify the slope and a point on the line when a linear equation is in this form.


rearrange9


Notice that when we have a linear equation in slope-intercept form, the slope is the number in front of the x, and the y-intercept is the number by itself. This is the exact information that we wanted.

For example, consider the line y = 3x + 2. The number in front of x is 3, so the slope is 3, and the number by itself is 2, so the y-intercept is (0, 2). We can use this information to plot another point and graph the line.


rearrange5


Rearranging Linear Equations Before Graphing

As we've seen, it's very easy to identify the slope and a point from the slope-intercept form of a line. However, we aren't always given a linear equation in this form. When we get a linear equation that is not in slope-intercept form, we can manipulate it, or rearrange it, to get it in slope-intercept form. Then we can identify the slope and y-intercept, and use those to graph the line. Any of the following things can be done to rearrange an equation without changing the solution of the equation:

  1. Add or subtract the same term from both sides.
  2. Multiply or divide both sides by the same non-zero term
  3. Simplify either side
  4. Interchange sides

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