Linear Relationship Rules

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Linear Rules or Functions are mathematical algebra equations which tell us how to get the output Y-values for a given set of input X-values.

The Rule tells us the “Relationship” between all of the x and y values.

If we graph any of these input and output (x,y) values, a straight line will be created.

We need to be able to find the Rule for Sets of (x,y) pairs, and for any straight line graphs we have drawn, so that:

1) We can find the “Y” values for very large x-values, without having to draw a giant graph and manually read them off

2) We can find the “Y” values for very small x-values, like x = 0.001, which would be way too small to try and read off a normal sized graph

3) We can type the rule into a Computer Graphing Application, and have the computer draw the graph for us

In relation to point 3) above, nobody draws graphs in the real world any more, they are all drawn by computer apps.

If you would like to try out an Online Graph Maker, then try the ones out at these two links:

http://webgraphing.com/plotting_basic.jsp#

http://rechneronline.de/function-graphs/

Real World Straight Line Graphs

If you would like to see where straight lines are used in the real World, then have a look at our lessons about this at the links below:

http://passyworldofmathematics.com/real-world-line-graphs/

and

http://passyworldofmathematics.com/straight-line-graphs/

Types of Linear Relationship Rules

There are basically 3 types of Linear Rules:

1) Simple Addition or Subtraction

Eg. y = x +2, y = x -5, y = x + 7, y = x -3 etc

2) Simple Multiplication or Division

Eg. y = 2x, y = -5x, y = x/2, y = -x/5 etc

3) Combination Rule using y = mx + c

Eg. y = 2x + 3, y = -3x + 1, y = 4x -2, y = -x + 7 etc

When we are given a Table of Input and Output x and y values, we check for:

1) Simple Addition or Subtraction

and if this does not work out then check for

2) Simple Multiplication or Division

and if this does not work out then check for

3) Combination Rule using y = mx + c

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Think of this as being like a Doctor in a Hospital Emergency Ward checking an incoming patient’s injuries.

They might check for :

1) Do they just have some cuts and bleeding

2) Do they just have broken bones

3) Do they have a combination of injuries

Just like the Doctor case, if we have situation 3), involving a combination of issues, then this situation will require the most work.

Simple Addition Rule

In this situation the difference between the y and x values will be the same number every time as we work down our set of (x,y) values.

If we cannot identify this relationship by simply looking at the numbers, then we can mathematically check for it.

We check by adding an extra “y-x” column to our given table of input “x” and output “y” values, and then doing the math.

In every case the y-x difference is always the same number value
of positive “1”.

The Linear Rule is y = x + 1

Check:

y = x + 1 (sub in our x,y is 0,1 value)

1 = 0 + 1

1 = 1

Our y = x + 1 rule works, and so our y = x + 1 answer is correct!

Zero Addition Rule

If the X-Y difference check produces zeroes, then we have the y=x rule.

This relationship should be obvious by just looking at the x and y values and seeing that they are the same.

We do not really need to do the Y-X column maths.

Simple Subtraction Rule

This is the exact same approach as the Simple Addition Rule, but our y-x values all come out the same NEGATIVE NUMBER each time.

In every case the y-x difference is always the same number value of NEGATIVE 2 OR “-2”.

The Linear Rule is y = x + -2

which we write as y = x – 2

Check:

y = x – 2 (sub in our x,y is 0,-2 value)

-2 = 0 – 2

-2 = -2

Our y = x – 2 rule works, and so our y = x – 2 answer is correct!

If we do not find a simple Addition or Subtraction Rule that works for our (x,y) points, then we need to move on to checking for a Simple Multiplication Relationship.

Simple Multiplication Rule

If there is a specific number that we can multiply each x value by, and this produces the y-value, then we have a Multiplication Relationship.

In this situation y divided by x for all values will be the same number every time, EXCEPT FOR X = 0

(It is impossible to divide any number by zero, and so we have written “–” in our table).

If we cannot identify this relationship by simply looking at the numbers, then we can mathematically check for it.

We can check for a “Multiplication Relationship” by adding an extra “y/x” column to our given table of input “x” and output “y” values, and then doing the math.

In every case y/x is always the same number value of positive “3”.

The Linear Rule is y = 3x

Check:

y = 3x (sub in our x,y is 1,3 value)

3 = (3)(1)

3 = 3

Our y = 3x rule works, and so our y = 3x answer is correct!

Simple Multiplication – Negative Values

Note that y/x can come out a negative number, such as -2.

If this happens the Linear Rule is y = -2x

Simple Multiplication – y/x value is 1

The y/x value can sometimes turn out as “1”.

If this happens the Linear Rule is y = 1x

which we write as:

y = x

Simple Division Rule

If our y/x values from a Simple Multiplication checking table turn out to be all the same fraction value, then we have a Simple Division Linear Rule.

The same y/x value we get each time is 1/4 .

Our Linear Rule is: y = 1/4 x

We can write this as: y = x/4

Combination Rule: y = mx + c

If we cannot identify a simple Addition, Subtraction, Multiplication, or Division relationship, then there is probably a “Combination Rule” involved.

Eg. If our y-x values, and our y / x values, do NOT give us a common number answer pattern, then we probably have a more complicated “Combination Rule” involved.

Combination Rules are more complicated, and involve both Multiplying and Adding to make a rule in the form:

y = mx + c or y = mx – c

where m = gradient, and c = y-intercept

Note that Americans, and some other countries, call the combination rule

y = mx + b and y = mx – b,

Combination Rule – Working Out Steps

There are four steps, as follows:

Step 1 involves working out the Change in x and y values.

Step 2 involves Calculating the Gradient or Slope as:

m = Change in Y / Change in X

If we have a Graph given to us, instead of an x,y Values Table, then we need to find the gradient “m” directly from the graph.

Now let’s look at Steps 3 and 4, where we find the “C” value for y = mx + c

People have a lot of trouble with this step, because it looks like there is too much working out to do.

Another way to approach Step 3 is to realize that we have half the rule already, because we know m = 2.

This means the Y values are all 2x + ?

If we now go back and visually inspect the X and Y-values Table, knowing that we need to double each “x” value and then also do something else,

we might be able to figure out what that something else is.

Once we figure out this something else is “adding 1” then we know that C = + 1, and the full rule is y = 2x + 1.

Once we have completed Steps 1 to 3, we know the number values for “m” and “c”, so all we need to do is put these values into y = mx + c

As we discussed above, once we know the gradient “m” we can look at the table values and should be able to figure out “c”.

We combine Steps 3 and 4 together into a single step:

Video About Finding y = mx + c Rules

Finding y = mx + c rules is not easy for beginners, and it might take a while until you fully figure out what is going on.

Watching the following video will help you better undertand what needs to be done for these types of questions.

Finding Rules for Line Graphs

When we have a Line Graph, the procedure is the same, except that our (x,y) points are on the line, rather than being in a values table.

The Procedure is:

Find the Gradient “m” between any two points on the line using Rise / Run

(The points are best located where the line crosses the corners of a set of four grid squares, so that we have whole number values for Rise and Run).

We then put our number value for “m” into y = mx + c, and use a set of x,y point values from the line to work out the value of “c”.

Finding Rule for Graphs – Example One

In the example below, we find the Gradient slope, and then from there on we have half the rule done.

All we need to do is work out what else is done to “x” to get the “Y” value, and this will tell us what “c” is.

Gradient “m” is always the multiply part of the rule, and “c” is always the add on or subtract part of the rule.

Finding Rule for Graphs – Example Two

In this next example, we do the same procedure:

First use “m” = Rise / Run to get the first part of the rule.

Then pick any point from the line, multiply the “x” coordinate by the “m” value, and then figure out what we need to add on, (or subtract away), to get the Y-Value.

The above example shows that we can use the Combination Rule y = mx + c for any type of straight line.

The Example 2 line was actually a simple “Addition Rule” relationship.

Even though we went through the more complicated “Combination Rule” steps, we still got to the correct answer.

Blank X-Y Grid

Here is a blank X-Y Grid you can print out and use for drawing question lines onto and working out rules.

Linear Rule Generator

The Maths Warehouse website has a great free java app on their page where we can move around the given straight line, and the appp works out the y = mx + c (as y = mx + b) rule for us.

It can be found on the following web page:

http://www.mathwarehouse.com/algebra/linear_equation/slope-intercept-form.php

Linear Relationship Rules

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