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# Linear Rules

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

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### Linear Rules

1. 1. Linear Relationship Rules Image Modified from Original at www.chiropractor.com
2. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 22. -3 3 5 -5
23. 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. 24. -3 3 5 -5
25. 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. 26. Blank X-Y Grid -3 3 5 -5
27. 27. http://passyworldofmathematics.com/