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© Boardworks Ltd 20091 of 27 Equations 1.Equations 2.Using inverse operations 3.Solving equations by transforming both sides 4.Solving an equation with unknowns on both sides

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© Boardworks Ltd 20092 of 25 © Boardworks Ltd 20092 of 27 Equations An equation links an algebraic expression and a number, or two algebraic expressions with an equals sign. For example: x + 7 = 13 is an equation. In an equation the unknown usually has a particular value. Finding the value of the unknown is called solving the equation. x + 7 = 13 x = 6 When we solve an equation we always line up the equals signs.

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© Boardworks Ltd 20093 of 25 © Boardworks Ltd 20093 of 27 Using inverse operations In algebra, letter symbols represent numbers. Rules that apply to numbers in arithmetic apply to letter symbols in algebra. In arithmetic, if 3 + 7 = 10, we can use inverse operations to write: 10 – 7 = 3and10 – 3 = 7 In algebra, if a + b = 10, we can use inverse operations to write: 10 – b = a and 10 – a = b a = 10 – b and b = 10 – a or

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© Boardworks Ltd 20094 of 25 © Boardworks Ltd 20094 of 27 b 12 = a Using inverse operations In arithmetic, if 3 × 4 = 12, we can use inverse operations to write: 12 ÷ 4 = 3and12 ÷ 3 = 4 In algebra, if ab = 12, we can use inverse operations to write: and or 12 b = a a = b a = b

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© Boardworks Ltd 20095 of 25 © Boardworks Ltd 20095 of 27 Using inverse operations to solve equations We can use inverse operations to solve simple equations. For example: x + 5 = 13 x = 13 – 5 x = 8 Always check the solution to an equation by substituting the solution back into the original equation. If we substitute x = 8 back into x + 5 = 13 we have 8 + 5 = 13

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© Boardworks Ltd 20096 of 25 © Boardworks Ltd 20096 of 27 Using inverse operations to solve equations Solve the following equations using inverse operations. 5 x = 45 x = 45 ÷ 5 x = 9 Check: 5 × 9 = 45 17 – x = 6 17 = 6 + x 17 – 6 = x Check: 17 – 11 = 6 11 = x x = 11 We always write the letter before the equals sign.

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© Boardworks Ltd 20097 of 25 © Boardworks Ltd 20097 of 27 Using inverse operations to solve equations Solve the following equations using inverse operations. x = 3 × 7 x = 21 Check: 3 x – 4 = 14 3 x = 14 + 4 3 x = 18 Check: 3 × 6 – 4 = 14 x = 18 ÷ 3 x = 6 = 3 7 x 7 21

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© Boardworks Ltd 20098 of 25 © Boardworks Ltd 20098 of 27 4 Solving equations by transforming both sides Solve this equation by transforming both sides in the same way: Add 1 to both sides. Multiply both sides by 4. m = 12 We can check the solution by substituting it back into the original equation: 12 ÷ 4 – 1 = 2 – 1 = 2 m 4 = 3 m +1 ×4

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© Boardworks Ltd 20099 of 25 © Boardworks Ltd 20099 of 27 Solving an equation with unknowns on both sides Let’s solve this equation by transforming both sides of the equation in the same way. 3 n – 11 = 2 n – 3 Start by writing the equation down. −2n−2n −2 n Subtract 2 n from both sides. n – 11 = –3 Always line up the equals signs. +11 Add 11 to both sides. n = 8 This is the solution. We can check the solution by substituting it back into the original equation: 3 8 – 11 =2 8 – 3

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