A linear function of x is a function of the form y = ax + b, where a
and b are constants. It's called a "linear" function because a
plot of the x and y coordinates for which this equation is true takes the shape
of a straight line.
Here's an example of a table of x and y values similar to
ones in (my son) Matthew's 6th grade math book:
The student is supposed to see a pattern, and fill in the blanks, then write
an equation for calculating the values in the table.
Here is the procedure for solving this type of problem.
- Start by assuming the relationship between x and y is linear. That
means whenever x increases by some specific amount (the x-increment), y will
increase by some other specific amount (the y-increment). The
y-increment depends only on the x-increment, not the value of x.
- Look for a basic x-increment. The first two x values differ by
3. The next two differ by 3. The next difference is 9, and the
final difference is 6. These are all multiples of 3, so the basic
x-increment is 3.
- Look for the ratio of the y-increment to the x-increment. Do this
using two rows of the table that have both x and y values shown.
Calculate the x-increment for these two rows by subtracting the x
values. Calculate the y-increment by subtracting the two y
values. Divide the y-increment by the x-increment to get the ratio of
the y-increment to the x-increment. In the example above, using the
first two rows, the x-increment is 10 - 7, or 3. The y-increment is 20
- 13, or 7. The ratio of the y-increment to the x-increment is
7/3. This means when the x-increment is 3, the y-increment is 7, which
is the basic y-increment.
- Check to see if the x values are multiples of the x-increment by dividing
the lowest x value by the basic x-increment, and looking at the
remainder. In our example, the lowest x value is 7. 7 divided by
the basic x-increment, 3, is 2 with a remainder of 1. I'll call this
- At this point, you are almost ready to calculate the y-value for any
x-value. Here's the procedure:
- Start with the x-value.
- Subtract the x-constant.
- Divide by the basic x-increment.
- Multiply by the basic y-increment.
- Add the y-constant.
- This gives you the y-value.
In our example, let's start with the first x-value, 7.
Subtracting the x-constant, 1, leaves 6.
Dividing by the basic x-increment, 3, gives 2.
Multiplying by the basic y-increment, 7, gives 14.
Now add the y-constant -- wait a minute! What's the y-constant?
The next step answers that question:
- To find the y-constant, use a row of the table that has both an
"x" and "y" value filled in. Now do the procedure,
above, through step "e". Subtract the result from the actual
y-value to get the y-constant. If the result of step e is bigger than
the actual y-value, then the y-constant is a negative number. In our
example, the result of step "e" was 14, and the actual y-value is
13 so the y-constant is -1.
So far I've taught you how to get the y-value if you know the x-value.
Suppose it's the other way around. What do you do if you know the
y-value, but not the x-value? It's simple. Just do the same
procedure, just switching the x's and y's.
- Here's how to find the x-value if you know the y-value:
- Start with the y-value.
- Subtract the y-constant.
- Divide by the basic y-increment.
- Multiply by the basic x-increment.
- Add the x-constant.
- This gives you the x-value.
In our example, let's start with the first y-value, 13.
Subtract the y-constant, -1 (remember subtracting a negative is really
adding) giving 14.
Divide by the basic y-increment, 7, giving 2.
Multiply by the basic x-increment, 3, giving 6.
Add the x-constant, 1, giving 7, which is the right answer!
- Now to write the formula, it's simple:
y = y-increment/x-increment * (x - x-constant) + y-constant
In our example, the formula is this:
y = 7 / 3 * (x - 1) - 1
Do you have any questions about this procedure? Send me
Related pages in this website
Matrix Math -- a way of solving much more
complicated linear systems involving several variables. This section
includes determinants, the Reduced Row Echelon method, also called
Gauss-Jordan Elimination and Cramer's rule.
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