How to find the Linear Rule for a Straight Line Graph or a Table of (x,y) values.

1. Linear Relationship
Rules
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2. 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

3. 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

4. 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

5. When we do y – x
for all of our (x,y)
pairs, we get the
same answer of “0”
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

6. When we do y – x
for all of our (x,y)
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

7. When we do y – x
for all of our (x,y)
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

8. When we do y / x
for all of our (x,y)
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

9. When we do y / x
for all of our (x,y)
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

10. When we do y – x
for all of our (x,y)
pairs, we get the
same answer of “0”
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

11. When we do y – x
for all of our (x,y)
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

12. 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

13. 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

14. 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

15. 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.

16. 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

17. -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

18. 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

19. 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

20. 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

21. -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.

22. -3 3
5
-5

23. -3 3
5
-5
Gradient m = 4/2 = 2
y = 2 x + ?
Use (x,y) = (2,6)
6 = 2 (2) + ?
6 = 4 + ?
The Rule is :
y = 2x + 2
Rise = 4
Run = 2

24. -3 3
5
-5

25. -3 3
5
-5
Rise = 2
Run = 2
Gradient m = 2/2 = 1
y = 1 x + ?
Use (x,y) = (0,1)
1 = 1 (0) + ?
1 = 0 + ?
The Rule is :
y = 1x + 1 y = x + 1

26. Blank X-Y Grid
-3 3
5
-5

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