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Equations 1. 2. 3. 4. Equations Using inverse operations Solving equations by transforming both sides Solving an equation with unknowns on both sides 1 of 27 © Boardworks Ltd 2009 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. 2 of 27 25 © Boardworks Ltd 2009 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 = 3 and 10 – 3 = 7 In algebra, if a + b = 10, we can use inverse operations to write: or 3 of 27 25 10 – b = a and 10 – a = b a = 10 – b and b = 10 – a © Boardworks Ltd 2009 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 = a b or 4 of 27 25 a = 12 b and 12 = b a and b = 12 a © Boardworks Ltd 2009 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 5 of 27 25 © Boardworks Ltd 2009 Using inverse operations to solve equations Solve the following equations using inverse operations. 17 – x = 6 5x = 45 x = 45 ÷ 5 17 = 6 + x 17 – 6 = x x=9 11 = x Check: 5 × 9 = 45 6 of 27 25 x = 11 We always write the letter before the equals sign. Check: 17 – 11 = 6 © Boardworks Ltd 2009 Using inverse operations to solve equations Solve the following equations using inverse operations. x 7 =3 x=3×7 x = 21 Check: 21 7 =3 3x – 4 = 14 3x = 14 + 4 3x = 18 x = 18 ÷ 3 x=6 Check: 3 × 6 – 4 = 14 7 of 27 25 © Boardworks Ltd 2009 Solving equations by transforming both sides Solve this equation by transforming both sides in the same way: m –1=2 4 +1 +1 Add 1 to both sides. m =3 4 ×4 ×4 Multiply both sides by 4. m = 12 We can check the solution by substituting it back into the original equation: 12 ÷ 4 – 1 = 2 8 of 27 25 © Boardworks Ltd 2009 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 Subtract 2n from both sides. −2n n – 11 = –3 +11 +11 n = 8 Always line up the equals signs. Add 11 to both sides. This is the solution. We can check the solution by substituting it back into the original equation: 3 8 – 11 = 2 8 – 3 9 of 27 25 © Boardworks Ltd 2009