Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Solving Equations: Multiple-step solutions Listen & Learn Number Sense and Algebra Solving Equations: Multiple-step solutions Listen & Learn PRESENTED BY ALGESTAR Mathematics, Grade 9 [Title] Solving Equations: Multiple-step solutions Introduction • Welcome to today’s topic • Parts of Listen & Learn Presentation, questions, Q&A • Housekeeping NOT the Chat Room Your questions Satisfaction Meter 1 Solving Equations: Multiple-step solutions Listen & Learn [Title] Solving Equations: Multiple-step solutions 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. [Title] Solving Equations: Multiple-step solutions Agenda • Background and importance • Definitions and terms • Back to basics • Multiple-step solutions: Multiple terms Brackets Fractions 2 Solving Equations: Multiple-step solutions Listen & Learn [Title] Solving Equations: Multiple-step solutions Agenda • Background and importance • Definitions and terms • Back to basics • Multiple-step solutions: Multiple terms Brackets Fractions [Title] Solving Equations: Multiple-step solutions ALGEBRA • Branch of mathematics that studies structure, relation and quantity • Uses letters or symbols to represent numbers and mathematical relations Figure 1: René Descartes. • “Solving” algebraic equations means determining the actual values the letters represent Portrait by Frans Hals, 1648. 3 Solving Equations: Multiple-step solutions Listen & Learn [Title] Solving Equations: Multiple-step solutions Importance Elementary algebra leads to Number theory Geometry Trigonometry Abstract algebra Algebra is used in the fields of Architecture Engineering Graphic arts Business/entrepreneurship [Title] Solving Equations: Multiple-step solutions Agenda • Background and importance • Definitions and terms • Back to basics • Multiple-step solutions: Multiple terms Brackets Fractions 4 Solving Equations: Multiple-step solutions Listen & Learn [Title] Solving Equations: Multiple-step solutions Definitions Variable Term Term that contains a letter (variable) Example: 7x, –4y, z Constant Term Term that contains ONLY a number Example: 9, –2 [Title] Solving Equations: Multiple-step solutions Definitions Coefficient The factor by which a variable is multiplied Example: –7 x coefficient variable variable term 5 Solving Equations: Multiple-step solutions Listen & Learn [Title] Solving Equations: Multiple-step solutions Definitions BEDMAS A mnemonic (memory helper) used to help with remembering the correct order of operations Brackets, Exponents, Division, Multiplication, Addition, Subtraction [Title] Solving Equations: Multiple-step solutions 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 Solving Equations: Multiple-step solutions Listen & Learn [Title] Solving Equations: Multiple-step solutions 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? [Title] Solving Equations: Multiple-step solutions 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 Solving Equations: Multiple-step solutions Listen & Learn [Title] Solving Equations: Multiple-step solutions Variable term The variable term in the expression –8x + 3 is a) 3 b) –8x [Title] Solving Equations: Multiple-step solutions Variable term The variable term in the expression –8x + 3 is a) 3 b) –8x This term contains the letter x (variable). 8 Solving Equations: Multiple-step solutions Listen & Learn [Title] Solving Equations: Multiple-step solutions Agenda • Background and importance • Definitions and terms • Back to basics • Multiple-step solutions: Multiple terms Brackets Fractions [Title] Solving Equations: Multiple-step solutions 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 Solving Equations: Multiple-step solutions Listen & Learn [Title] Solving Equations: Multiple-step solutions 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 – 2 + 2 = 14 + 2 x = 16 (the 2 was regrouped with the 14) [Title] Solving Equations: Multiple-step solutions 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 Solving Equations: Multiple-step solutions Listen & Learn [Title] Solving Equations: Multiple-step solutions Back to basics Solve: –2w = 42 a) w = 21 b) w = 44 c) w = –21 d) w = 40 [Title] Solving Equations: Multiple-step solutions Back to basics Solve: –2w = 42 a) w = 21 b) w = 44 c) w = –21 d) w = 40 11 Solving Equations: Multiple-step solutions Listen & Learn [Title] Solving Equations: Multiple-step solutions Back to basics Solve: –2w = 42 w is multiplied by –2 Inverse of multiplying is dividing on the other side [Title] Solving Equations: Multiple-step solutions Agenda • Background and importance • Definition and terms • Back to basics • Multiple-step solutions: Multiple terms Brackets Fractions 12 Solving Equations: Multiple-step solutions Listen & Learn [Title] Solving Equations: Multiple-step solutions 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 [Title] Solving Equations: Multiple-step solutions 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 + 6 – 6 = 18 – 6 –4x = 12 13 Solving Equations: Multiple-step solutions Listen & Learn [Title] Solving Equations: Multiple-step solutions Multiple terms x is still being multiplied by –4 –4x = 12 Move the –4 by dividing on the other side (BEDMAS) [Title] Solving Equations: Multiple-step solutions 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 Solving Equations: Multiple-step solutions Listen & Learn [Title] Solving Equations: Multiple-step solutions 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) [Title] Solving Equations: Multiple-step solutions Multiple terms Solve: 3 + 3x – 10 = 5x + 31 Use inverse operations to regroup 3 – 10 – 31 = 5x – 3x Simplify in each group –38 = 2x Divide by the coefficient of x –19 = x 15 Solving Equations: Multiple-step solutions Listen & Learn [Title] Solving Equations: Multiple-step solutions Multiple terms Solve: 4x – 7 + 9x = –12 – 2x Is this a correct next step? 4x + 9x – 2x = –12 + 7 a) Yes b) No [Title] Solving Equations: Multiple-step solutions Multiple terms Solve: 4x – 7 + 9x = –12 – 2x Is this a correct next step? 4x + 9x – 2x = –12 + 7 a) Yes b) No 16 Solving Equations: Multiple-step solutions Listen & Learn [Title] Solving Equations: Multiple-step solutions 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 = –12 + 7 15x = –5 [Title] Solving Equations: Multiple-step solutions Brackets Solve: –3(5 – 6a) = 39 Simplify the equation first by removing the brackets using the Distributive Property (–3 x 5) + (–3 x –6a) = 39 –15 + 18a = 39 17 Solving Equations: Multiple-step solutions Listen & Learn [Title] Solving Equations: Multiple-step solutions Brackets Now regroup terms by using inverse operations in reverse –15 – 39 = –18a Simplify –54 = –18a Divide by the coefficient of ‘a’ 3=a [Title] Solving Equations: Multiple-step solutions Brackets Given –15 + 18a = 39 Can I regroup as: 18a = 39 + 15? a) Yes b) No 18 Solving Equations: Multiple-step solutions Listen & Learn [Title] Solving Equations: Multiple-step solutions Brackets Given –15 + 18a = 39 Can I regroup as: 18a = 39 + 15? a) Yes b) No [Title] Solving Equations: Multiple-step solutions Brackets Can I regroup as: 18a = 39 + 15? 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 Solving Equations: Multiple-step solutions Listen & Learn [Title] Solving Equations: Multiple-step solutions 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 [Title] Solving Equations: Multiple-step solutions Brackets Solve: 3(2h – 5) + 4(3h + 2) = 11 Distribute 6h – 15 + 12h + 8 = 11 Regroup 6h + 12h = 11 + 15 – 8 Simplify 18h = 18 Divide by coefficient of variable (h) h=1 20 Solving Equations: Multiple-step solutions Listen & Learn [Title] Solving Equations: Multiple-step solutions Fractions Solve: Contains a single fraction Regroup by moving +6 first [Title] Solving Equations: Multiple-step solutions Fractions Move 2 from the denominator by multiplying it on the other side 21 Solving Equations: Multiple-step solutions Listen & Learn [Title] Solving Equations: Multiple-step solutions Fractions Solve: More than one fraction Clear out the fractions first Make sure each term has a denominator Choose a common denominator [Title] Solving Equations: Multiple-step solutions Fractions Common denominator = 10 Multiply each term by the common denominator and reduce the fractions 22 Solving Equations: Multiple-step solutions Listen & Learn [Title] Solving Equations: Multiple-step solutions Fractions Continue solving as a multiple-term equation Solve: 4x + 20 = 5 Regroup 20 with 5 by subtracting 4x = 5 – 20 4x = –15 Divide by coefficient of x [Title] Solving Equations: Multiple-step solutions Fractions The solution of is 105. a) True b) False 23 Solving Equations: Multiple-step solutions Listen & Learn [Title] Solving Equations: Multiple-step solutions Fractions The solution of is 105. a) True b) False [Title] Solving Equations: Multiple-step solutions Fractions Solve Regroup 4 with 19 by subtracting Regroup 7 with 15 by multiplying -q = 15 x 7 -q = 105 24 Solving Equations: Multiple-step solutions Listen & Learn [Title] Solving Equations: Multiple-step solutions Fractions In the final solution, the variable cannot have a negative coefficient. Divide by –1 (coefficient of q) q = –105 [Title] Solving Equations: Multiple-step solutions Resources Purplemath www.purplemath.com/modules/solvelin .htm Algebra.help www.algebrahelp.com/lessons Math.com homeworkhelp www.math.com/school/subject2/lessons /S2U3L1GL.html 25