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Solving Equations A-REI.B.3:Solving equations with a variable on one side, using inverse operations.

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**Solving Equations Vocabulary**

Equation – a mathematical sentence (statement) that shows two expressions are equivalent (equal). Solve – to isolate the variable. Solution – the value that makes the equation a true statement. Inverse operations – operations that “undo” each other; addition and subtraction, multiplication and division.

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What will happen if you add or subtract an equal amount of weight on both sides of the scales? Solving equations is like balancing scales, we must always keep the sides equal.

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Addition Property of Equality – states you can add the same amount to both sides of an equation and the equation remains true = = = 9 ? true Subtraction Property of Equality – states you can subtract the same amount from both sides of an equation and the equation remains true = – 3 = 11 – = 8 ? true

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**Solving Equations by Multiplication or Division**

Multiplication Property of Equality – states you can multiply the same amount on both sides of an equation and the equation remains true. 4 · 3 = 12 2 · 4 · 3 = 12 · 2 24 = 24 Division Property of Equality – states you can divide the same amount on both sides of an equation and the equation remains true = 6 2

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**To solve two-step equations, undo the operations by working backwards.**

Recall the order of operations as you answer these questions. Dividing by 2 Subtracting 3 Example: 𝑥 2 −3=−7 Ask yourself, What is the first thing we are doing to x? What is the second thing? To undo these steps, do the opposite operations in opposite order.

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**To solve two-step equations, undo the operations by working backwards.**

Build is our “Do” Reverse is our “Undo” ★ No, you are not allowed to solve problems like that! ★ Instead show solve more like this but don’t skip in between steps.

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**To solve two-step equations, undo the operations by working backwards.**

Build is our “Do” Reverse is our “Undo” ★ No, you are not allowed to solve problems like that! ★ Instead show solve more like this but don’t skip in between steps.

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**How do I “undo” a fraction?**

3 4 The fraction can be separated into two operations; 1.What operation is being performed by the numerator? 2. What is being performed by the denominator? In order to ‘undo’ a fraction to isolate the variable, break the fraction down into 2 parts DO UNDO •3 ÷ 4 Multiplying by 3 Dividing by 4 • 4 ÷ 3

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**Use a DO-UNDO chart as a shortcut to answering the questions**

Use a DO-UNDO chart as a shortcut to answering the questions. In the table, write the opposite operations in the opposite order 𝑥 2 −3=−7 Draw “the river” Add 3 to both sides Simplify Multiply both sides by 2 Check your answer DO UNDO ÷2 -3 Follow the steps in the ‘undo’ column to isolate the variable. +3 • 2 𝑥 = −8 𝑥 2 =−4 2• • 2 −8 2 −3=−7 −4−3=−7 −7=−7

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**1) Solve 2𝑥 − 1 = −3 +1 +1 2𝑥 =−2 2 2 𝑥 = -1 D U Draw “the river”**

· 2 - 1 + 1 ÷ 2 2𝑥 =−2 𝑥 = -1 Draw “the river” Add 1 to both sides Simplify Divide both sides by 2 Check your answer 2(−1) − 1 = −3 −2 – 1 = −3 −3 = −3

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**𝑥 3 −4=8 2) Solve +4 +4 3· · 3 𝑥 = 36 𝑥 3 =12 36 3 −4=8 12 – 4 = 8 8=8**

D U 2) Solve 𝑥 3 −4=8 ÷ 3 - 4 + 4 · 3 3· · 3 𝑥 = 36 𝑥 3 =12 Draw “the river” Add 4 to both sides Simplify Multiply both sides by 3 Check your answer 36 3 −4=8 12 – 4 = 8 8=8

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3) Solve 3𝑦 – 1 = 8 𝑦 = 3 𝑦 = −3 𝑦 = 7 3 𝑦 =− 7 3 Answer Now

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**3) Solve 𝑑 = 10 𝑑 – 4 = 6 +4 +4 𝑑−4 2 =3 2 𝑑−4 2 =3 2 D U**

- 4 ÷ 2 · 2 + 4 2 𝑑−4 2 =3 2 Draw “the river” Clear the fraction - Multiply both sides by 2 Simplify Add 4 to both sides Check your answer 𝑑 – 4 = 6 𝑑 = 10 10−4 2 =3 6 2 =3 3=3

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4) Solve 𝑑+1 3 =−6 𝑑 = −7 𝑑 = −19 𝑑 = −17 𝑑 = 17 Answer Now

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**5) Solve 3− 𝑎 7 =−2 −3 −3 − 𝑎 7 =−5 𝑎 = 35 −7 − 𝑎 7 =−5 −7 3− 35 7 =−2**

D U 5) Solve 3− 𝑎 7 =−2 ÷ -7 + 3 - 3 · -7 Do/Undo Chart – This one is tricky! Remember to always use the sign in front of the number. − −3 − 𝑎 7 =−5 𝑎 = 35 Draw “the river” Subtract 3 from both sides Simplify Clear the fraction – Multiply both sides by -7 Check your answer −7 − 𝑎 7 =−5 −7 3− 35 7 =−2 3−5=−2 −2=−2

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6) Solve 5𝑧 + 16 = 51 𝑧 = −35 𝑧 = −7 𝑧 = 35 𝑧 = 7 Answer Now

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**7) Solve 3 5 𝑥+1 =4 −1 −1 3 5 𝑥 =3 5• •5 3𝑥 = 15 3 3 𝑥 = 5 D U**

· 3 ÷ 5 + 1 - 1 · 5 ÷ 3 − −1 Draw “the river” Subtract 1 from both sides Simplify Clear the fraction -Multiply both sides by 5 Divide both sides by 3 Check your answer 3 5 𝑥 =3 5• •5 3𝑥 = 15 𝑥 = 5 =4 3+1=4 4=4

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