Slide 1 / 116 Slide 2 / 116 New Jersey Center for Teaching and Learning Solving Equations Progressive Mathematics Initiative Algebra I This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course 2012-11-16 materials to parents, students and others. www.njctl.org Click to go to website: www.njctl.org Slide 3 / 116 Slide 4 / 116 Table of Contents Inverse Operations Click on a topic to One Step Equations go to that section. Inverse Operations Two Step Equations Multi-Step Equations Variables on Both Sides More Equations Return to Table of Transforming Formulas Contents These sections are a repeat from 8th grade Common Core. Use with your students as necessary. Slide 5 / 116 Slide 6 / 116 Equations can also be used to state the equality of two expressions containing one or more variables. What is an equation? An equation is a mathematical statement, in symbols, In real numbers we can say, for example, that for any that two things are exactly the same (or equivalent). given value of x it is true that Equations are written with an equal sign, as in 4x + 1 = 14 - 1 2+3=5 If x = 3, then 9-2=7 4(3) + 1 = 14 - 1 12 + 1 = 13 13 = 13
Slide 7 / 116 Slide 8 / 116 When defining your variables, remember... An equation can be compared to a balanced scale. Letters from the beginning of the alphabet like a , b , c ... often denote constants in the context of the discussion at hand. Both sides need to contain the same While letters from end of the alphabet, like x , y , z ..., are usually quantity in order for it to reserved for the variables, a convention initiated by Descartes. be "balanced". Try It! Write an equation with a variable and have a classmate identify the variable and its value. Slide 9 / 116 Slide 10 / 116 Why are we Solving Equations? For example, 20 + 30 = 50 represents an equation because both sides simplify to 50. First we evaluated expressions where we were given the value of the variable and had to find what the expression 20 + 30 = 50 simplified to. 50 = 50 Now, we are told what it simplifies to and we need to find the Any of the numerical values in the equation can be value of the variable. represented by a variable. When solving equations, the goal is to isolate the variable on Examples: one side of the equation in order to determine its value (the value that makes the equation true). 20 + c = 50 x + 30 = 50 20 + 30 = y Slide 11 / 116 Slide 12 / 116 There are four properties of equality that we will use to solve In order to solve an equation containing a variable, you need equations. They are as follows: to use inverse (opposite/undoing) operations on both sides of the equation. Addition Property If a=b, then a+c=b+c for all real numbers a, b, and c. The same number can be added to each side of the equation without Let's review the inverses of each operation: changing the solution of the equation. Subtraction Property Addition Subtraction If a=b, then a-c=b-c for all real numbers a, b, and c. The same number can be subtracted from each side of the equation Multiplication Division without changing the solution of the equation. Multiplication Property If a=b, and c=0, then ac=bc for all real numbers ab, b, and c. Each side of an equation can be multiplied by the same nonzero number without changing the solution of the equation. Division Property If a=b, and c=0, then a/c=b/c for all real numbers ab, b, and c. Each side of an equation can be divided by the same nonzero number without changing the solution of the equation.
Slide 13 / 116 Slide 14 / 116 For each equation, write the inverse operation needed to solve for the variable. a.) y +7 = 14 subtract 7 b.) a - 21 = 10 add 21 move move c.) 5s = 25 divide by 5 move d.) x = 5 multiply by 12 move 12 Slide 15 / 116 Slide 16 / 116 Think about this... Think about this... To solve c - 3 = 12 In the expression Which method is better? Why? To which does the "-" belong? Kendra Ted Does it belong to the x? The 5? Both? Added 3 to each side of Subtracted 12 from each side, The answer is that there is one negative so it is used once with the equation then added 15. either the variable or the 5. Generally, we assign it to the 5 to avoid creating a negative variable. c - 3 = 12 c - 3 = 12 Touch to reveal answer +3 +3 -12 -12 So: c = 15 c - 15 = 0 +15 +15 c = 15 Slide 17 / 116 Slide 18 / 116
Slide 19 / 116 Slide 20 / 116 To solve equations, you must work backwards through the order of operations to find the value of the variable. Remember to use inverse operations in order to isolate the variable on one side of the equation. One Step Equations Whatever you do to one side of an equation, you MUST do to the other side! Return to Table of Contents Slide 21 / 116 Slide 22 / 116 One Step Equations Examples: y + 9 = 16 Solve each equation then click the box to see work & solution. - 9 -9 The inverse of adding 9 is subtracting 9 x - 8 = -2 2 = x - 6 y = 7 +8 +8 +6 +6 click to show click to show x = 6 inverse operation 8 = x inverse operation 6m = 72 6 6 The inverse of multiplying by 6 is dividing by 6 7 = x + 3 x + 2 = -14 m = 12 -3 -3 -2 -2 click to show 4 = x inverse operation click to show x = -16 inverse operation Remember - whatever you do to one side of an equation, you MUST do to the other!!! 15 = x + 17 x + 5 = 3 -5 -5 -17 -17 click to show click to show x = -2 -2 = x inverse operation inverse operation Slide 23 / 116 Slide 24 / 116 One Step Equations 3x = 15 x (2) 3 3 (2) = 10 click to show 2 x = 5 inverse operation x = 20 click to show inverse operation -4x = -12 -4 -4 click to show x = 3 inverse operation x (-6) = 36 (-6) -6 x = -216 -25 = 5x click to show 5 5 inverse operation click to show -5 = x inverse operation
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Slide 31 / 116 Slide 32 / 116 Slide 33 / 116 Slide 34 / 116 Sometimes it takes more than one step to solve an equation. Remember that to solve equations, you must work backwards through the order of operations to find the value of the variable. Two-Step Equations This means that you undo in the opposite order (PEMDAS): 1st: Addition & Subtraction 2nd: Multiplication & Division 3rd: Exponents 4th: Parentheses Return to Whatever you do to one side of an equation, you MUST do to Table of the other side! Contents Slide 35 / 116 Slide 36 / 116 Examples: Two Step Equations 3x + 4 = 10 Solve each equation then click the box to see work & solution. - 4 - 4 Undo addition first 3x = 6 3 3 Undo multiplication second 6-7x = 83 3x + 10 = 46 -4x - 3 = 25 x = 2 - 10 -10 +3 +3 -6 -6 3x = 36 -4x = 28 -7x = 77 -4 -4 -7 -7 3 3 -4y - 11 = -23 x = -7 x = 12 x = -11 + 11 +11 Undo subtraction first -4y = -12 -4 -4 Undo multiplication second y = 3 -2x + 3 = -1 9 + 2x = 23 8 - 2x = -8 -9 -9 - 3 -3 -8 -8 Remember - whatever you do to one side 2x = 14 -2x = -4 -2x = -16 of an equation, you MUST do to the other!!! 2 2 -2 -2 -2 -2 x = 7 x = 2 x = 8
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Slide 49 / 116 Slide 50 / 116 Steps for Solving Multiple Step Equations As equations become more complex, you should: 1. Simplify each side of the equation. (Combining like terms and the distributive property) Multi-Step Equations 2. Use inverse operations to solve the equation. Remember, whatever you do to one side of an equation, you MUST do to the other side! Return to Table of Contents Slide 51 / 116 Slide 52 / 116 Examples: Now try an example. Each term is infinitely cloned so you can pull them down as you solve. -15 = -2x - 9 + 4x -15 = 2x - 9 Combine Like Terms +9 +9 Undo Subtraction first -6 = 2x -7x + 3 + 6x = -6 2 2 Undo Multiplication second -3 = x 7x - 3x - 8 = 24 4x - 8 = 24 Combine Like Terms x = 9 + 8 +8 Undo Subtraction first 4x = 32 4 4 Undo Multiplication second x = 8 answer Slide 53 / 116 Slide 54 / 116 Now try another example. Each term is infinitely cloned so you can pull them down as you solve. Always check to see that both sides of the equation are 6x + x = 44 - 5 simplified before you begin solving the equation. Sometimes, you need to use the distributive property in order to simplify part of the equation. x = -9 answer
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