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1 Number Sense and Algebra Solving Equations: Multiple-step PRESENTED BY ALGESTAR Mathematics, Grade 9 Introduction Welcome to today s topic Parts of Presentation, questions, Q&A Housekeeping NOT the Chat Room Your questions Satisfaction Meter 1

2 What you will learn At the end of this lesson, you will be able to solve equations that require multiple steps to reach a final solution. Equations can have multiple terms, brackets, or fractions in them. Agenda Background and importance Definitions and terms Back to basics Multiple-step : Multiple terms Brackets Fractions 2

3 Agenda Background and importance Definitions and terms Back to basics Multiple-step : Multiple terms Brackets Fractions ALGEBRA Figure 1: René Descartes. Portrait by Frans Hals, Branch of mathematics that studies structure, relation and quantity Uses letters or symbols to represent numbers and mathematical relations Solving algebraic equations means determining the actual values the letters represent 3

4 Importance Elementary algebra leads to Number theory Geometry Trigonometry Abstract algebra Algebra is used in the fields of Architecture Engineering Graphic arts Business/entrepreneurship Agenda Background and importance Definitions and terms Back to basics Multiple-step : Multiple terms Brackets Fractions 4

5 Definitions Variable Term Term that contains a letter (variable) Example: 7x, 4y, z Constant Term Term that contains ONLY a number Example: 9, 2 Definitions Coefficient The factor by which a variable is multiplied Example: coefficient 7 x variable variable term 5

6 Definitions BEDMAS A mnemonic (memory helper) used to help with remembering the correct order of operations Brackets, Exponents, Division, Multiplication, Addition, Subtraction Definitions Distributive Property States that when a sum is multiplied by a number, each value in the sum is multiplied separately, and then the products are added. Example: 3(p 4) = 3(p +( 4)) = (3 x p) + (3 x 4) = 3p 12 6

7 Terms Simplify means to add/subtract the coefficients of like terms in an expression. e.g., 5x + 3x = 8x Solve means to find the value of the variable that makes the equation true. e.g., if x 3 = 4, what is x? Terms Inverse Operations are mathematical processes that are the opposite of each other. e.g., Addition is the inverse operation of subtraction. e.g., Division is the inverse operation of multiplication. 7

8 Variable term The variable term in the expression 8x + 3 is a) 3 b) 8x Variable term The variable term in the expression 8x + 3 is a) 3 b) 8x This term contains the letter x (variable). 8

9 Agenda Background and importance Definitions and terms Back to basics Multiple-step : Multiple terms Brackets Fractions Back to basics Solving simple equations: First identify the operation acting on the variable. Then isolate the variable by undoing the operation (using the inverse operation). NOTE: It will look like terms are moving, or being regrouped within the equation. 9

10 Back to basics Solving simple equations: Example: x 2 = 14 2 is being subtracted from x Inverse of subtracting is adding on the other side of the equation x = x = 16 (the 2 was regrouped with the 14) Back to basics Solving simple equations: Example: x is divided by 3 Inverse of dividing is multiplying on the other side of the equation. x = 8 x 3 x = 24 10

11 Back to basics Solve: 2w = 42 a) b) c) d) w = 21 w = 44 w = 21 w = 40 Back to basics Solve: 2w = 42 a) b) c) d) w = 21 w = 44 w = 21 w = 40 11

12 Back to basics Solve: 2w = 42 w is multiplied by 2 Inverse of multiplying is dividing on the other side Agenda Background and importance Definition and terms Back to basics Multiple-step : Multiple terms Brackets Fractions 12

13 Multiple terms Solve: 4x + 6 = 18 Equation involves more than one operation We need to undo EACH operation being applied to the variable in the reverse order to BEDMAS, until the variable is isolated Multiple terms Look for terms that are added or subtracted first (BEDMAS) 4x + 6 = 18 The inverse of adding is subtracting. So, move (regroup) the 6 to the right side of the equation by subtracting it from 18. 4x = x = 12 13

14 Multiple terms x is still being multiplied by 4 4x = 12 Move the 4 by dividing on the other side (BEDMAS) Multiple terms Solve: 3 + 3x 10 = 5x + 31 Decide which terms (i.e., variables on one side, constants on the other) will go on which side of the equals sign Easier to go left to right through the question, moving terms (regrouping like terms together) as necessary. 14

15 Multiple terms Solve: If we choose to have constants on the left side: 3 + 3x 10 = 5x + 31 (these terms don t move) 3 + 3x 10 = 5x + 31 (these terms need to be regrouped) Multiple terms Solve: 3 + 3x 10 = 5x + 31 Use inverse operations to regroup = 5x 3x Simplify in each group 38 = 2x Divide by the coefficient of x 19 = x 15

16 Multiple terms Solve: 4x 7 + 9x = 12 2x Is this a correct next step? 4x + 9x 2x = a) Yes b) No Multiple terms Solve: 4x 7 + 9x = 12 2x Is this a correct next step? 4x + 9x 2x = a) Yes b) No 16

17 Multiple terms Solve: 4x 7 + 9x = 12 2x When you regroup 2x, which is on the right side of the equation, it becomes +2x on the left side. 4x + 9x + 2x = x = 5 Brackets Solve: 3(5 6a) = 39 Simplify the equation first by removing the brackets using the Distributive Property ( 3 x 5) + ( 3 x 6a) = a = 39 17

18 Brackets Now regroup terms by using inverse operations in reverse = 18a Simplify 54 = 18a Divide by the coefficient of a 3 = a Brackets Given a = 39 Can I regroup as: 18a = ? a) Yes b) No 18

19 Brackets Given a = 39 Can I regroup as: 18a = ? a) Yes b) No Brackets Can I regroup as: 18a = ? When regrouping terms, the variable term (a) can be on the left or the right side of the equation. One guideline for deciding is to let the first term stay where it is, and move the other terms accordingly. 19

20 Brackets In this case, keeping the variable term on the left means needing to move only one term instead of two. The final solution will still be the same in both cases, although when solving the equation, the signs for the terms in each step will be opposite. 18a = 54 a = 3 Brackets Solve: 3(2h 5) + 4(3h + 2) = 11 Distribute 6h h + 8 = 11 Regroup 6h + 12h = Simplify 18h = 18 Divide by coefficient of variable (h) h = 1 20

21 Fractions Solve: Contains a single fraction Regroup by moving +6 first Fractions Move 2 from the denominator by multiplying it on the other side 21

22 Fractions Solve: More than one fraction Clear out the fractions first Make sure each term has a denominator Choose a common denominator Fractions Common denominator = 10 Multiply each term by the common denominator and reduce the fractions 22

23 Fractions Continue solving as a multiple-term equation Solve: 4x + 20 = 5 Regroup 20 with 5 by subtracting 4x = x = 15 Divide by coefficient of x Fractions The solution of is 105. a) True b) False 23

24 Fractions The solution of is 105. a) True b) False Fractions Solve Regroup 4 with 19 by subtracting Regroup 7 with 15 by multiplying -q = 15 x 7 -q =

25 Fractions In the final solution, the variable cannot have a negative coefficient. Divide by 1 (coefficient of q) q = 105 Resources Purplemath Algebra.help Math.com homeworkhelp /S2U3L1GL.html 25