Lesson 2.1 Solving Equations w/Justification Concept: Solving Equations EQ: How do we justify how we solve equations? REI. 1 Vocabulary: Properties of Equality Properties of Operation Justify 1

Solve the equations below, provide an explanation for your steps. 2

Properties of Equality 2.1.1: Properties of Equality 3 PropertyIn symbolsExample Reflexive property of equality a = a2=2 Symmetric property of equality If a = b, then b = a. x = 3 3 = x Transitive property of equality If a = b and b = c, then a = c. x = 2, y = 2, x = y Addition property of equality If a = b, then a + c = b + c. x – 4 = 3 x – = x = 7

Properties of Equality, continued 2.1.1: Properties of Equality 4 PropertyIn symbolsExamples Subtraction property of equality If a = b, then a – c = b – c. x + 2 =5 x + 2 – 2 = 5 – 2 x = 3 Multiplication property of equality If a = b and c ≠ 0, then a c = b c.x=15 Division property of equality If a = b and c ≠ 0, then a ÷ c = b ÷ c. 4x = 16 x = 4

Properties of Equality, continued 2.1.1: Properties of Equality 5 PropertyIn symbolsExamples Substitution property of equality If a = b, then b may be substituted for a in any expression containing a. x = 3, then 2x = 2(3) = 6

Properties of Operations 2.1.1: Properties of Equality 6 PropertyGeneral ruleSpecific example Commutative property of addition a + b = b + a3 + 8 = Associative property of addition (a + b) + c = a + (b + c)(3 + 8) + 2 = 3 + (8 + 2) Commutative property of multiplication a b = b a3 8 = 8 3 Associative property of multiplication (a b) c = a (b c)(3 8) 2 = 3 (8 2) Distributive property of multiplication over addition a (b + c) = a b + a c3 (8 + 2) =

Guided Practice Example 1 Which property of equality is missing in the steps to solve the equation –7x + 22 = 50? 2.1.1: Properties of Equality 7 EquationSteps –7x + 22 = 50Original equation –7x = 28 x = –4Division property of equality

Guided Practice: Example 1, continued 1. Observe the differences between the original equation and the next equation in the sequence. What has changed? Notice that 22 has been taken away from both expressions, –7x + 22 and : Properties of Equality 8

Guided Practice: Example 1, continued 2. Refer to the table of Properties of Equality. The subtraction property of equality tells us that when we subtract a number from both sides of the equation, the expressions remain equal. The missing step is “Subtraction property of equality.” 2.1.1: Properties of Equality 9 ✔

Guided Practice: Example 1, continued 2.1.1: Properties of Equality 10

2.1.1: Properties of Equality 11 EquationSteps Original equation Addition property of equality –x = 42 x = –42Division property of equality

2.1.1: Properties of Equality 12

Guided Practice: Example 2, continued In order to move to the next step, the division of 6 has been undone. The inverse operation of the division of 6 is the multiplication of 6. The result of multiplying by 6 is –x and the result of multiplying 7 by 6 is 42. This matches the next step in the sequence : Properties of Equality 13

Guided Practice: Example 2, continued 2. Refer to the table of Properties of Equality. The multiplication property of equality tells us that when we multiply both sides of the equation by a number, the expressions remain equal. The missing step is “Multiplication property of equality.” 2.1.1: Properties of Equality 14 ✔

Guided Practice: Example 2, continued 2.1.1: Properties of Equality 15

Guided Practice: Example 3 What equation is missing based on the steps? 1. Observe the 3 rd and 5 th equations. 2. Read the 4 th step. 3. Fill in the missing equation : Properties of Equality 16

You Try… Identify the property of equality that justifies each missing step or equation EquationSteps 9 + x = 17 Original Equation x = 8 EquationSteps 7(2x + 1) = 49Original Equation 14x + 7 = 49 14x = 42 Subtraction Property of Equality x = 3

5. Solve the equation that follows. Justify each step in your process using the properties of equality. Be sure to include the properties of operations, if used. 8(2x – 1) = 56 18

Summary… Identify the property represented below. 1. x -3 = 6 x = A = B, B = C, then A = C Solve the problem below justifying each step using the properties of equality. 3. 2x – 9 = 1 19

Solving Equations with the Variable in Both Expressions of the Equation 1. Move the variable to solve for to the left of the equal sign. 2. Move all other terms to the right of the equal sign. 3. Combine like terms on each side of the equal sign. 4. Now solve for the variable and simplify. 5. Substitute the solution into the original equation and check your work. 20

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