Linear Rules

Linear Relationship
Rules
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Linear Relationship
Rules
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

Linear Relationship
Rules
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
X Y
-2 -1
-1 0
0 1
1 2
2 3

When we do y – x
for all of our (x,y)
pairs, we get the
same answer of “1”
The rule is :
y = x + 1
X Y Y - X
-2 -1 -1 - -2 = 1
-1 0 0 - -1 = 1
0 1 1 – 0 = 1
1 2 2 – 1 = 1
2 3 3 – 2 = 1
To check for “Addition Rule”: Add an extra column and work out y-x values

When we do y – x
pairs, we get the
The rule is :
y = x + 0 which is
y = x
X Y Y – X
-2 -2 -2 - -2 = 0
-1 -1 -1 - -1 = 0
0 0 0 – 0 = 0
1 1 1 – 1 = 0
2 2 2 – 2 = 0
To check for “Addition Rule”: Add an extra column and work out y-x values

When we do y – x
pairs, we get the
same answer of “-2”
The rule is :
y = x - 2
X Y Y - X
-1 -3 -3 - -1 = -2
0 -2 -2 - 0 = -2
1 -1 -1 – 1 = -2
2 0 0 – 2 = -2
3 1 1 – 3 = -2
Same as “Addition Rule”: Add an extra column and work out y-x values

When we do y – x
pairs, we get all
Different answers.
The rule is NOT
Addition or
Subtraction, so we
now check for the
Multiplication Rule.
X Y Y - X
-1 -3 -3 - -1 = -2
0 0 0 - 0 = 0
1 3 3 – 1 = 2
2 6 6 – 2 = 4
3 9 9 – 3 = 6
If we work out y-x values, but do not get “same answers”, then we need
to move on and try the y / x “Multiplication Rule” checking routine

When we do y / x
pairs, we get the
same answers.
The rule is :
y = 3x
X Y Y / X
-1 -3 -3 / -1 = 3
0 0 0 / 0 = --
1 3 3 / 1 = 3
2 6 6 / 2 = 3
3 9 9 / 3 = 3
We work out results by doing y / x “Multiplication Rule” checking

When we do y / x
pairs, we get the
same answer of “-2”.
The rule is :
y = -2x
X Y Y / X
-1 2 2 / -1 = -2
0 0 0 / 0 = --
1 -2 -2 / 1 = -2
2 -4 -4 / 2 = -2
3 -6 -6 / 3 = -2
We work out results by doing y / x “Multiplication Rule” checking

When we do y – x
pairs, we get the
The rule is :
y = 1x which is
y = x
X Y Y / X
-2 -2 -2 / -2 = 1
-1 -1 -1 / -1 = 1
0 0 0 / 0 = --
1 1 1 / 1 = 1
2 2 2 / 2 = 1
Check for “Multiplication Rule”: Add extra column and work out y/x values

When we do y – x
pairs, we get the
same answer of “-2”
The rule is :
y = ¼ x which is:
y = x / 4
X Y Y / X
-4 -1 -1 / -4 = ¼
0 0 0 / 0 = --
4 1 1 / 1 = ¼
8 2 2 / 2 = ¼
12 3 3 / 3 = ¼
Same as “Multiplication Rule”: Add extra column and work out y/x values

X Y Y – X
0 1 1 - 0 = 1
1 3 3 - 1 = 2
2 5 5 - 2 = 3
3 7 7 - 3 = 4
4 9 9 - 4 = 5
If our y-x values, and our y / x values, do not give us the same answer
pattern, then we have a more complicated “Combination Rule” involved.
X Y Y / X
0 1 1 / 0 = --
1 3 3 / 1 = 3
2 5 5 / 2 = 2.5
3 7 7 / 3 = 2.3
4 9 9 / 4 = 2.2

X Y
0 1
1 3
2 5
3 7
4 9
If our y-x values, and our y / x values, do not give us the same answer
pattern, then we 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
m = slope, and c = y-intercept

X Y
0 1
1 3
2 5
3 7
4 9
Working out a “Combination Rule” involves several working out steps
Step 1) Work out Change in x and
Change in y patterns.
Step 2) m = Chg in y / Chg in x
Step 3) Use “m” and any x,y pair
to work out what “c” is.
Step 4) Rule is y = mx + c
after we substitute in the step 2
and 3 “m” and “c” answers

X Y
0 1
1 3
2 5
3 7
4 9
Step 1 involves working out the Change in x and y values as shown here
+ 2
+ 2
+ 2
+ 2+ 1
+ 1
+ 1
+ 1
The Change
in x-values
is positive 1
each time.
The Change
in y-values
is positive 2
each time.

X Y
0 1
1 3
2 5
3 7
4 9
Calculate the Gradient Slope as: m = Change in Y / Change in X
+ 2
+ 2
+ 2
+ 2+ 1
+ 1
+ 1
+ 1
m = Change in Y / Change in X
m = +2 / + 1
m = 2

-3 3
5
-5
If we have a Graph
instead of an x,y Values
Table, then use:
Gradient m = Rise/Run
Gradient m = 4/2 = 2
m = 2
Rise = 4
Run = 2

X Y
0 1
1 3
2 5
3 7
4 9
Use “m” with any x,y value to find out what the y-intercept “c” is
+ 2
+ 2
+ 2
+ 2+ 1
+ 1
+ 1
+ 1
m = 2
Choose an “easy”
x,y value like 1,3
and along with m=2
and substitute into :
y = mx + c
3 = (2)(1) + c
So “c” must be:
c = 1

X Y
0 1
1 3
2 5
3 7
4 9
Substitute the “m” and “c” values into y = mx + c Rule Equation
+ 2
+ 2
+ 2
+ 2+ 1
+ 1
+ 1
+ 1
m = 2 c = 1 Substitute C = 1
with m=2 into :
y = mx + c
y = 2x + 1
The “Combination”
Rule is:
y = 2x + 1

X Y
0 1
1 3
2 5
3 7
4 9
Use “m” with any x,y value to find out what the y-intercept “c” is
+ 2
+ 2
+ 2
+ 2+ 1
+ 1
+ 1
+ 1
m = 2
y = mx + c
We know m=2 so
y = 2x + ?
Look at the Table 1,3
3 = (2)(1) + ? So
Y = 2x + 1

-3 3
5
-5
Steps are these:
Find gradient “m”
Put “m” into y = mx + c
Write y = mx + ?
Use an (x,y) point from
the Line to figure out “c”
Write final answer.

-3 3
5
-5
y = 2 x + ?
Use (x,y) = (2,6)
6 = 2 (2) + ?
6 = 4 + ?
The Rule is :
y = 2x + 2
Rise = 4
Run = 2

-3 3
5
-5
Rise = 2
Run = 2
y = 1 x + ?
Use (x,y) = (0,1)
1 = 1 (0) + ?
1 = 0 + ?
The Rule is :
y = 1x + 1 y = x + 1

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Linear Rules

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Linear Rules