2.2 Linear Equations from Data

You will not always be given the equation for a linear function. Instead you will have available some information about it and, from that data, you will have to work out the equation.

The most common case is to have two data points, (x1,y1) and (x2,y2) which satisfy the equation. Then, since you are working with the slope-intercept form of linear equation, you should first work out the slope, b, as follows:

Next, substitute one data point, say (x1,y1) into the equation to find the intercept, a, as follows:

which can then be solved for the intercept as it is the only unknown.

Example 2.2.3

Suppose that in the poster printing example above you were only given the facts that, to make 1,000 posters, it would cost $190 and, to make 1,500 posters, it would cost $245. The information could be organized by thinking of the points as  being in a table or on  a graph as follows:

Then to find the slope:

To find the y-intercept:


and write the whole equation as follows:

Knowledge Check 2.2

There are two temperature scales in North America: Fahrenheit and Celsius. They are related by a linear equation. The temperature of boiling water is 212° Fahrenheit and 100° Celsius. The temperature of freezing water is 32° Fahrenheit and 0° Celsius.

Let,

F = temperature on the Fahrenheit scale (the y).
C = temperature on the Celsius scale (the x).

Then find the equation for F in terms of C (i.e., ) by using the following steps:

  1. Choose one point as (x1,y1) another as (x2,y2) in the formula,
  2. Since you now know b, substitute b and one of the known points in 
    and solve for a.
  3. Write the final equation and find the Fahrenheit temperature equivalent to 20 degrees Celsius (approximately room temperature).

 

 

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