A Line with the Right Slope

Joan used the following directions to graph a line identical to the green line we had drawn:

1. Find the y-intercept of the line.
2. From this point, draw a line with the green line's slope.

We need to turn these directions into an equation. Making an equation will actually be easier if we follow the directions backward:

1. Draw a line with the right slope.
2. Slide the line up the y-axis until it has the right y-intercept.

Drawing a Line with the Right Slope

Let's try to write an equation for a line with a slope of 3. Since lots of different lines have slopes of 3, we'll try to keep things simple by writing an equation for the line that goes through the origin. Notice that a yellow fly is sitting on this line at point (1, 3).

We need to find a connection between the x-coordinates and the y-coordinates of the points on this line.

We can start by making a table with the x-coordinates in one column and the y-coordinates in the other column.

0	0
2/3	2
1	3
4/3	4

Do you see a pattern in these numbers? Look at the point that the fly is sitting on. When x = 1, y = 3. Since 3 * 1 = 3, we could try guessing that 3 * x = y. Does this formula work for the other numbers in the table?

3 * 0 = 0

3 * 2/3 = 2

3 * 4/3 = 4

It looks as if 3 * x = y for all the pairs of x- and y-coordinates in our table. This makes sense, because the line had a slope of 3 to begin with. We should expect to see a 3 in our final equation.