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ALGEBRA EQUATIONS ► Goals for solving equations – Isolate the variable, and use the inverse operations to undo the operation performed on the variable. ► Equation – a mathematical statement that two expressions are equal. ► Solution of an Equation – a value of the variable that makes the equation true.

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ADDITION AND SUBTRACTION PROPERTIES OF EQUALITY ► Addition Property of Equality – You can add the same number to both sides of an equation, and the statement will still be true. Ex: x – 10 = 4 + 10 +10 + 10 +10 x = 14 x = 14 Subtraction Property of Equality – You can subtract the same number from both sides of an equation, and the statement will still be true. Ex: x + 7 = 9 - 7 -7 - 7 -7 x = 2 x = 2

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Solving Equations by Adding the Opposite ► Remember that subtracting is the same as adding the opposite. When solving equations, you will sometimes find it easier to add an opposite to both sides instead of subtracting. Ex: -8 + b = 2 + 8 +8 + 8 +8 b = 10 b = 10

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MULTIPLICATION AND DIVISION PROPERTIES OF EQUALITY ► Multiplication Property of Equality – You can multiply both sides of an equation by the same number, and the statement will still be true. Ex: 4 = k/5 (5)4 = (5)k/5 (5)4 = (5)k/5 20 = k 20 = k Division Property of Equality – You can divide both sides of an equation by the same nonzero number, and the statement will still be true. Ex: 42 = 7b 42 = 7b 42 = 7b 7 7 7 7 6 = b 6 = b

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Solving Equations that Contain Fractions ► Remember that dividing is the same as multiplying by the reciprocal. When solving equations that have a fraction multiplied to the variable, you can isolate the variable by multiplying both sides by the reciprocal. Ex: ½ v = 35 (2) ½ v = (2) 35 (2) ½ v = (2) 35 v = 70 v = 70

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Steps for Solving for a Variable 1. Locate the variable you are asked to solve for in the equation. 2. Identify the operations on this variable and the order in which they are applied. 3. Use inverse operations to “undo” the operations and isolate the variable.

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Solving Two-Step Equations

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Goal: To isolate the variable ► 1. Undo any addition or subtraction that occurs on the side by the variable. ► 2. Undo any multiplication or division that occurs on the side by the variable. ► 3. Check your solution to make sure that the equation is true.

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Special Cases 1. Remember to use the distributive property as needed and to combine like terms. 2. To solve equations that involve fractions, you may use the Multiplication Property of Equality. Multiply both sides of the equation by the least common denominator to clear fractions.

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Example ► 18 = 4a + 10 -10 -10 -10 -10 8 = 4a 8 = 4a 4 4 4 4 2 = a 2 = a Check: 4 * 2 + 10 = 18

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Example ► 5t – 2 = -32 + 2 + 2 + 2 + 2 5t = -30 5t = -30 5 5 5 5 t = -6 t = -6 Check: 5 * -6 – 2 = -32

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Example ► y/4 – 3/4 = 5/4 (multiply by the LCD to clear fractions) (multiply by the LCD to clear fractions) 4(y/4 – 3/4) = 4(5/4) 4(y/4) – 4(3/4) = 4(5/4) y – 3 = 5 y – 3 = 5 + 3 +3 + 3 +3 y = 8 y = 8 Check: 8/4 – 3/4 = 5/4

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Example ► 2/3r + 3/4 = 7/12 (the LCD of 3, 4, and 12 is 12) (the LCD of 3, 4, and 12 is 12) 12(2/3r + 3/4) = 12(7/12) Multiplication Prop. Of Eq. 12(2/3r + 3/4) = 12(7/12) Multiplication Prop. Of Eq. 12(2/3r) + 12(3/4) = 12(7/12) Distributive Prop. 12(2/3r) + 12(3/4) = 12(7/12) Distributive Prop. 8r + 9 = 7 Simplify 8r + 9 = 7 Simplify - 9 - 9 - 9 - 9 8r = -2 8r = -2 8 8 8 8 r = -1/4 r = -1/4 Check: 2/3(-1/4) + 3/4 = 7/12

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Example ► 8x – 21 - 5x = -15 8x + (-5x) + (-21) = -15 Commutative Property 8x + (-5x) + (-21) = -15 Commutative Property 3x – 21 = -15 Combine like terms 3x – 21 = -15 Combine like terms + 21 +21 Addition property of equality + 21 +21 Addition property of equality 3x = 6 3x = 6 3 3 Division Property of Equality 3 3 Division Property of Equality x = 2 x = 2 Check: 8(2) – 21 – 5(2) = -15 16 - 21 – 10 = -15 16 - 21 – 10 = -15

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Example ► 10y – (4y + 8) = -20 10y + (-1)(4y + 8) = -20 To subtract, add the opposite 10y + (-1)(4y + 8) = -20 To subtract, add the opposite 10y + (-1)(4y) + (-1)(8) = -20 Distributive Property 10y + (-1)(4y) + (-1)(8) = -20 Distributive Property 10y + -4y + -8 = -20 Simplify 10y + -4y + -8 = -20 Simplify 6y + -8 = -20 Combine like terms 6y + -8 = -20 Combine like terms +8 +8 Add the opposite of -8 +8 +8 Add the opposite of -8 6y = -12 6y = -12 6 6 Division Property of Equality 6 6 Division Property of Equality y = -2 y = -2 Check: 10(-2) – (4 * -2 + 8) -20 – 0 = -20 -20 – 0 = -20

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Example ► Sarah paid $15.95 to become a member at a gym. She then paid a monthly membership fee. Her total cost for 12 months was $735.95. How much was the monthly fee? ► Let f represent the monthly fee ► 15.95 + 12f = 735.95

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Lesson 2-4 Equations with Variables on Both Sides ► To solve an equation with variables on both sides first get the variable terms on one side. Then solve by undoing the operations performed on the variable in reverse order of operations. ► Identity – an equation that is true for all values of the variable. Ex: 2 + x = 2 + x ► Contradiction – an equation that is not true for any value of the variable. It has no solution. Ex: x = x + 3

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Sequence for Solving Equations 1. Distribute. 2. Simplify each side of the equation by combining like terms. 3. Move variables to one side of the equation. 4. Undo addition and subtraction. 5. Undo Multiplication and division. 6. CHECK.

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Tell which of the following is an identity. Explain your answer: A. 4(a+3)-6 = 3(a+3)-6 B. 8.3x-9+0.7x = 2 + 9x-11 An equation with variables can have… One solutionMany solutionsNo solution

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