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Graph a line 2x-3y=6

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Hi Katelyn,

The easiest way to do this is to change the equation so that it is in slope-intercept form: y = mx + b

where m = the slope (meaning the direction / angle of the line)

and      b = the y-intercept (the place where the line hits or intercepts the y axis)

Once you have the equation in this form, it will be a very simple drawing activity to graph it. 

So let's start by moving the equation around to get it in the form that we need it, slope-intercept form (where y is the only thing left on one side of the equation):

y = mx + b                                               2x - 3y = 6

                                                             -2x             -2x

                                                                   - 3y = -2x + 6

                                                                     /-3         /-3 

                                                                       y = 2/3x - 2    

(Notice that because we divided the whole equation by a negative 3, it did two things:

1) It cancelled out the -2 attached to the x and made a positive fraction, 2/3

2) It divided positive 6 by negative 3, leaving a negative number: -2 

This is a small but very important detail because it changes both the direction of the line and the place that it falls on the grid.)

Okay, now that we have our equation in slope intercept form as we figured out, it looks like this: 

                                                                    y = 2/3x - 2

                                                          where m = 2/3 and b = -2

                                          Now remember, m = slope, and b= y-intercept. 

The next part you have to remember is what the slope rules are.

                      slope is always rise/run                            rise      ↑      

                                                               over       --------------

                                                                                    run      →


So since the slope in this case is 2/3  (2 over 3), It will be easy to draw the line once we know where it goes. No matter where on the grid we end up putting it, each point will always be 2 segments up the y axis, and 3 segments over on the x axis (to the right). 2 segments up, and 3 segments over to the right. 2 segments up, and 3 segments over to the right. And it will be a straight line going through each of those points.  

Where do we place it on the graph, though? 

The answer is at the y-intercept, which in this case we determined is -2. The line will continue in both directions, because it doesn't actually start or end at the y axis. The idea with the equation of a line is that it continues indefinitely through space unless otherwise stated. But having the y-intercept DOES tell us where on the graph it crosses over the y axis. So we'll start there.      

(let's assume in this picture that:

the orange line is the x axis,

the thicker black numbers are the y axis,

and the o ' s are the points being graphed)

You'll have to use your imagination here to draw a straight-and-never ending line that connects each point on the graph. However, this is a basic summary of what it will look like:

- - - - - - - - - - - - - - - - - - -4- - - - - - - - - - - - - - - - - - - - - - - - - - - - - o - - - -

- - - - - - - - - - - - - - - - - - -3- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - - - - - - - - - - -2- - - - - - - - - - - - - - - - - - - o - - - - - - - - - - - - - -

- - - - - - - - - - - - - - - - - - -1- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

6 -5- - -4- - -3- - -2- - -1- - -l- - -1- - -2- - -o- - -4- - -5- - -6- - -7- - -8- - -9- -10- 

- - - - - - - - - - - - - - - - - - -1- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 

- - - - - - - - - - - - - - - - - - -o- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - - - - - - - - - -  3- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - -o - - - - - - - - -4- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - - - - - - - - - - -5- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

o - - - - - - - - - - - - - - - - -6- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

(Notice how the o that's on the y axis is below the x axis. This is because the y intercept was a negative number (-2). The negative numbers on the y axis start below the x axis. The point where the x and y axes meet is at 0, then each starts counting up when they've crossed over each other and continued on.)

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