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**Equations Equations Using inverse operations**

Solving equations by transforming both sides Solving an equation with unknowns on both sides

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**Boardworks KS3 Maths 2009 A4 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. Point out that all equations contain equals signs. In fact, the word equation starts with the same four letters as the word equals. x + 7 = 13 x = 6 When we solve an equation we always line up the equals signs.

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**Using inverse operations**

Boardworks KS3 Maths 2009 A4 Equations 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 = 10, we can use inverse operations to write: 10 – 7 = 3 and 10 – 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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**Using inverse operations**

Boardworks KS3 Maths 2009 A4 Equations Using inverse operations In arithmetic, if 3 × 4 = 12, we can use inverse operations to write: 12 ÷ 4 = 3 and 12 ÷ 3 = 4 In algebra, if ab = 12, we can use inverse operations to write: 12 b = a 12 a = b and b 12 = a a 12 = b or and

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**Using inverse operations to solve equations**

Boardworks KS3 Maths 2009 A4 Equations 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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**Using inverse operations to solve equations**

Boardworks KS3 Maths 2009 A4 Equations Using inverse operations to solve equations Solve the following equations using inverse operations. 5x = 45 17 – x = 6 x = 45 ÷ 5 17 = 6 + x x = 9 17 – 6 = x 11 = x Check: x = 11 5 × 9 = 45 We always write the letter before the equals sign. Check: 17 – 11 = 6

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**Using inverse operations to solve equations**

Boardworks KS3 Maths 2009 A4 Equations Using inverse operations to solve equations Solve the following equations using inverse operations. = 3 7 x 3x – 4 = 14 x = 3 × 7 3x = 3x = 18 x = 21 x = 18 ÷ 3 Check: x = 6 = 3 7 21 Check: 3 × 6 – 4 = 14

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**Solving equations by transforming both sides**

Boardworks KS3 Maths 2009 A4 Equations Solving equations by transforming both sides Solve this equation by transforming both sides in the same way: m – 1 = 2 4 +1 Add 1 to both sides. 4 = 3 m ×4 Multiply both sides by 4. Use this slide to go through as many examples as you need to ensure that pupils are confident with this written method for solving linear equations. Note that when we write this out the steps written in orange can be performed mentally. Every other step must be written on a new line with the equals signs lined up. Give pupils some written ‘What number am I thinking of?’ problems. Ask them to solve these by constructing an equation and solving it. Set pupils a variety of simple linear equations to solve. Start with equations that can be solved in a single step, for example 4a = 20, and progress to equations that require two steps, for example, 4b + 7 = 9. Include examples such as 6 = 2p – 8 and tell pupils to start by writing this as 2p – 8 = 6 (Remind pupils that these two are equivalent). m = 12 We can check the solution by substituting it back into the original equation: 12 ÷ 4 – 1 = 2

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**Solving an equation with unknowns on both sides**

Boardworks KS3 Maths 2009 A4 Equations Solving an equation with unknowns on both sides Let’s solve this equation by transforming both sides of the equation in the same way. 3n – 11 = 2n – 3 Start by writing the equation down. −2n −2n Subtract 2n from both sides. n – 11 = –3 Always line up the equals signs. +11 +11 Add 11 to both sides. Go through the steps shown to solve the equation step by step. Emphasize that we always do the same operation to both sides of the equation. Remind pupils that we want the equation in the form n = ‘a number’. Once we have a solution, demonstrate how we can check the solution by substituting it back into the original equation making sure that the left hand side of the equation is equal to the right hand side. 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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EXAMPLE 1 Solve a two-step equation Solve + 5 = 11. x 2 Write original equation. + 5 = x 2 11 + 5 – 5 = x 2 11 – 5 Subtract 5 from each side. = x 2 6 Simplify.

EXAMPLE 1 Solve a two-step equation Solve + 5 = 11. x 2 Write original equation. + 5 = x 2 11 + 5 – 5 = x 2 11 – 5 Subtract 5 from each side. = x 2 6 Simplify.

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