Equations

In some of the examples of the previous section we graphed lines using a point on the line and the slope of the line. In this section we'll review how to find the equations of lines.

Slope-Intercept Form

The graph of y = mx + b

is a line whose slope is m and whose y-intercept is (0,b).

Example 1

Find the equation of the line with slope of 2/3 with y-intercept of (0, 2).

Solution .

Using the formula y = mx + b, we get

If we make a graph of a line through (0, 2) with a run of 3 and a rise of 2, we get

This graph show that (3, 4) is on the graph. Note that when x = 3 is substituted into we get y = 4.

Example 2

Find the slope and y-intercept of 3x - 2y = 9.

Solution .

To put this into slope intercept form, solve for y.

Adding -3x to both sides, we get

Now dividing both sides by -2, we have

which is

The slope is 3/2 and the y-intercept is (0, -9/2)

Point-Slope Form

The point-slope form of a line passing through (x1, y1) with slope of m is

y - y1 = m(x - x1)

Example 3

Find the equation of the line through ( -2, 3) with slope of -5.

Solution .

Using y - y1 = m(x - x1) with x1 = -2, y1 = 3 and m = -5, we obtain

which becomes . Expanding and simplifying we get

As a check note that the slope of this line is -5 and that when you replace x by -2 you get a y of 3.

Example 4

Suppose a company had 4.8 million dollars of sales in the first quarter of the year and 5.6 million dollars of sales in the second quarter. Assuming a linear relationship between sales (y) in millions of dollars and time (x) in quarters, find y in terms of x.

Solution .

We are given two points (1, 4.8) and (2, 5.6). or .8 million dollars per quarter. Using x1 = 1 and y1 = 4.8, we get

which simplifies to

As a check putting x = 2 into this last equation yields y = 5.6.

Example 5

Make a table of values for y = 2x + 1 and y = 1.9x + 2 with x from 0 to 11 in steps of one.

Solution .

In the following table f(x) = 2x + 1 and g(x) = 1.9x + 2.

 x            f(x)         g(x)  

-------------------------------------

0            1.           2.         

1.000000000  3.000000000  3.900000000

2.000000000  5.000000000  5.800000000

3.000000000  7.000000000  7.700000000

4.000000000  9.000000000  9.600000000

5.000000000  11.00000000  11.50000000

6.000000000  13.00000000  13.40000000

7.000000000  15.00000000  15.30000000

8.000000000  17.00000000  17.20000000

9.000000000  19.00000000  19.10000000

10.00000000  21.00000000  21.00000000

11.00000000  23.00000000  22.90000000

Note that even though f(x) = 2x + 1 starts out smaller it eventually overtakes g(x) = 1.9x + 2 . What happens when both lines have the same slope?

In the following table, f(x) = 2x + 1 while h(x) = 2x +2.

 x            f(x)         h(x)  

-------------------------------------

0            1.           2.         

1.000000000  3.000000000  4.000000000

2.000000000  5.000000000  6.000000000

3.000000000  7.000000000  8.000000000

4.000000000  9.000000000  10.00000000

5.000000000  11.00000000  12.00000000

6.000000000  13.00000000  14.00000000

7.000000000  15.00000000  16.00000000

8.000000000  17.00000000  18.00000000

9.000000000  19.00000000  20.00000000

10.00000000  21.00000000  22.00000000

11.00000000  23.00000000  24.00000000

Will f(x) ever overtake h(x)?