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1 Topic 2.2.1 Properties of Equality

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2 Topic 2.2.1 Properties of Equality California Standard: 4.0 Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2 x – 5) + 4( x – 2) = 12. What it means for you: You’ll solve one-step equations using properties of equality. Key words: expression equation variable solve isolate equality

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3 Topic 2.2.1 Properties of Equality Now it’s time to use the material on expressions you learned in Section 2.1. In this Topic you’ll solve equations that involve addition, subtraction, multiplication, and division. An equation contains two expressions, with an equals sign in the middle to show that they’re equal. For example: 2 x – 3 = 4 x + 5

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4 Topic 2.2.1 An Equation Shows That Two Expressions are Equal Properties of Equality An equation is a way of stating that two expressions have the same value. The expression on the left-hand side... 24 – 9 = 15...has the same value as the expression on the right-hand side Some equations contain unknown quantities, or variables. The left-hand side... 2 x – 3 = 5...equals the right-hand side The value of x that satisfies the equation is called the solution (or root) of the equation. This equation contains only numbers — there are no unknowns:

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5 Topic 2.2.1 Addition and Subtraction in Equations Properties of Equality Subtraction Property of Equality For any real numbers a, b, and c, if a = b, then a – c = b – c. Finding the possible values of the variables in an equation is called solving the equation. Addition Property of Equality For any real numbers a, b, and c, if a = b, then a + c = b + c. These properties mean that adding or subtracting the same number on both sides of an equation will give you an equivalent equation. This may allow you to isolate the variable on one side of the equals sign.

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6 Topic 2.2.1 Example 1 Solution follows… Properties of Equality Solve x + 9 = 16. Solution x = 7 So subtract 9 from both sides to get x on its own. ( x + 9) – 9 = 16 – 9 x + 0 = 16 – 9 x = 16 – 9 You want x on its own, but here x has 9 added to it. x + 9 = 16 x + (9 – 9) = 16 – 9 Now simplify the equation to find x.

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7 Topic 2.2.1 Properties of Equality In the last Example, we solved x + 9 = 16 and found that x = 7 is the root of the equation. If x takes the value 7, then the equation is satisfied. If x takes any other value, then the equation is not satisfied. For example, if x = 6, then the left-hand side has the value 6 + 9 = 15, which does not equal the right-hand side, 16.

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8 Topic 2.2.1 Properties of Equality In other words, you just need to use the inverse operations. When you’re actually solving equations, you won’t need to go through all the stages each time — but it’s really important that you understand the theory of the properties of equality. If you have a “+ 9” that you don’t want, you can get rid of it by just subtracting 9 from both sides. If you have a “– 9” that you want to get rid of, you can just add 9 to both sides.

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9 Topic 2.2.1 Example 2 Solution follows… Properties of Equality Solve x + 10 = 12. Solution x + 10 = 12 x = 2 x = 12 – 10Subtract 10 from both sides Given equation

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10 Topic 2.2.1 Example 3 Solution follows… Properties of Equality Solve x – 7 = 8. Solution x – 7 = 8 x = 15 x = 8 + 7Add 7 to both sides Given equation

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11 Topic 2.2.1 Guided Practice Solution follows… Properties of Equality 1. x + 7 = 152. x + 2 = –8 3. 4. x – (–9) = –17 5. –9 + x = 106. x – 0.9 = 3.7 7. 8. –0.5 = x – 0.125 In Exercises 1–8, solve the equation for the unknown variable. x = 8 x = 19 x = –10 x = –26 x = 4.6 x = –0.375

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12 Topic 2.2.1 Multiplication and Division in Equations Properties of Equality Multiplication Property of Equality For any real numbers a, b, and c, if a = b, then a × c = b × c. These properties mean that multiplying or dividing by the same number on both sides of an equation will give you an equivalent equation. This can help you to isolate the variable and solve the equation. Division Property of Equality For any real numbers a, b, and c, such that c 0, if a = b, then.

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13 Topic 2.2.1 Multiply or Divide to Get the Variable on Its Own Properties of Equality the inverse operations. Once again, you just need to use the inverse operations. As with addition and subtraction, you can get the variable on its own by simply performing the inverse operation. If you have “× 3” on one side of the equation, you can get rid of that value by dividing both sides by 3. If you have a “÷ 3” that you want to get rid of, you can just multiply both sides by 3.

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14 Topic 2.2.1 Example 4 Solution follows… Solve 2 x = 18. Solution You want x on its own... but here you’ve got 2 x. x = 9 Divide both sides by 2 to get x on its own 1 x = 9 2 x = 18 Now simplify the equation to find x. Properties of Equality

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15 Topic 2.2.1 Example 5 Solution follows… Solve Solution m = 21 Multiply both sides by 3 to get m on its own Properties of Equality

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16 Topic 2.2.1 Example 6 Solution follows… Solve 4 x – 2(2 x – 1) = 2 – x + 3( x – 4). Solution 4 x – 2(2 x – 1) = 2 – x + 3( x – 4) Some equations are a bit more complicated. Take them step by step. 4 x – 4 x + 2 = 2 – x + 3 x – 12 2 = –10 + 2 x x = 6 2 x = 12 Clear out any grouping symbols Then combine like terms Given equation Properties of Equality

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17 Topic 2.2.1 Guided Practice Solution follows… Properties of Equality In Exercises 9–16, solve each equation for the unknown variable. 9. 4 x = 14410. –7 x = –7 13. –3 x + 4 = 1914. 4 – 2 x = 18 15. 3 x – 2( x – 1) = 2 x – 3( x – 4)16. 3 x – 4( x – 1) = 2( x + 9) – 5 x 11. 12. x = 36 x = 1 x = –28 x = –5 x = –7 x = 5 x = 7

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18 Topic 2.2.1 Independent Practice Solution follows… Properties of Equality Solve each of these equations: 1. –2(3 x – 5) + 3( x – 1) = –5 x = 4 2. 4(2 a + 1) – 5( a – 2) = 8 a = –2 3. 5(2 x – 1) – 4( x – 2) = –15 x = –3 4. 2(5 m + 7) – 3(3 m + 2) = 4 m 5. 4(5 x + 2) – 5(3 x + 1) = 2( x – 1)

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19 Topic 2.2.1 Independent Practice Solution follows… Properties of Equality 6. b – {3 – [ b – (2 – b ) + 4]} = –2(– b – 3) 7. 4[3 x – 2(3 x – 1) + 3(2 x – 1)] = 2[–2 x + 3( x – 1)] – (5 x – 1) 8. 30 – 3( m + 7) = –3(2 m + 27) 9. 8 x – 3(2 x – 3) = –4(2 – x ) + 3( x – 4) – 1 10. –5 x – [4 – (3 – x )] = –(4 x + 6) b = 7 m = –30 x = 6 Solve each of these equations:

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20 Topic 2.2.1 Independent Practice Solution follows… Properties of Equality In Exercises 11–17, solve the equations and check your solutions. You don’t need to show all your steps. 11. 4 t = 6012. x + 21 = 19 13. 14. 7 – y = –11 15. 16. 40 – x = 6 17. t = 15 x = –2 p = 8 y = 18 y = 28 x = 34 s = 16 18. Solve –(3 m – 8) = 12 – m m = – 2

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21 Topic 2.2.1 Independent Practice Solution follows… Properties of Equality 19. Denzel takes a two-part math test. In the first part he gets 49 points and in the second part he gets of the x points. If his overall grade for the test was 65, find the value of x. 20. Latoya takes 3 science tests. She scores 24%, 43%, and x % in the tests. Write an expression for her average percentage over the 3 tests. Her average percentage is 52. Calculate the value of x. Solve the following equations. Show all your steps and justify them by citing the relevant properties. 21. x + 8 = 1322. 11 + y = 15 23. y – 7 = 1924. 25. 4 m = 16 total points = ; x = 89 x = 5 y = 4 y = 26 x = 12 m = 4 ; x = 36

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22 Topic 2.2.1 Round Up Properties of Equality Solving an equation means isolating the variable. Anything you don’t want on one side of the equation can be “taken over to the other side” by using the inverse operation. You’re always aiming for an expression of the form: “ x =...” (or “ y =...,” etc.).

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