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Algebra Solving Linear Equations Copy into your notes What is an equation? An equation must have an = sign. Both sides are the same 3x + 9 = 15 Left hand side Right hand side To balance an equation: What you do to one side you must do to the other side. An equation is solved when you are have x on its own on one side e.g. x = 5 x 5 SLO To solve one step equations Copy into your notes Solving one step Equations Steps to solving equations. Step 1) Opposite on both sides Step 2) Cancel one side Step 3) Solve other side Check your answer Leave a couple centimetres above and below as there are other steps. We will cover this later. x + 9 = 15 _________ -9 -9 x = 6 First, put the opposite (- 9) on both sides. Second, cancel out the 9’s on the left side. Third, combine the right hand side. Check your answer: 6 + 9 = 15 K 7x = 3 x 7 7 K = 21 First, put the opposite on both sides. Second, cancel out one side. Third, combine the other side Check your answer: 21 ÷ 7 = 3 x+9=5 _________ –9 –9 x= -4 First, put the opposite (- 9) on both sides. Second, cancel out the 9’s on the left side. Third, combine the right hand side. Check your answer: -4 + 9 = 5 -3x =___ 15 ___ -3 -3 x = -5 First, put the opposite on both sides. Second, cancel out one side. Third, combine the other side. Check your answer Your Turn: Solve the following and show your workings Question x + 6 = 10 x–4=3 x+7=2 3x = -18 𝑋 3 =7 4x = 20 𝑋 4 = -2 Answer 4 7 -5 -6 21 5 -8 SLO To solve 2 step equations http://www.youtube.com/watch?v=juG-iIuTJQE&feature=relmfu (2 step equations) Examples of Two-Step Equations 2 Step Equations on a Balance 3x + 2 17 3x + 2 = 17 There are two operations to undo, but which comes first? Copy into your notes Rule for Solving 2 Step Equations 3x + 2 = 17 The operation to undo first is the one that is furthest away from x. In the above case we do the opposite of +2 first. Step 1) Opposite on both sides (furthest away first) Step 2) Cancel one side Step 3) Solve other side Step 4) Repeat until left with x on its own Check your answer Ex. 1: Solve 3x + 2 = 17 3x + 2 = 17 -2 -2 3x = 15 3 3 x=5 (Subtract 2 from both sides) (Cancel one side & solve other side) (Divide both sides by 3) (Cancel one side & solve other side) Check your answer: 3 x 5 + 2 = 17 correct Ex. 2: Solve 2x + 7 = 19 2x + 7 = 19 -7 -7 2x = 12 2 2 x=6 (Subtract 7 from both sides) (Cancel one side & solve other side) (Divide both sides by 2) (Cancel one side & solve other side) Check your answer: 2 x 6 + 7 = 19 correct Your Turn: Solve these and show your workings 1. 2x – 5 = 17 x = 11 2. 3y + 7 = 25 y=6 3. 5n – 2 = 38 n=8 4. 12b + 4 = 28 b=2 Questions to do from the books 2 step linear Achieve Gamma CAT 1.2 Merit P44 Ex4.01 P44 Ex4.02 P18 Q113 – 118, 127, 129 Excellence SLO To solve equations with x on both sides http://www.youtube.com/watch?v=f15zA0PhSek (x both sides) xxxxx xx 15 Write the above balance as an equation 5x = 2x + 15 What is the fist step to solving this equation? Take 2x What would you have left if you take 2x from both sides? 3x = 15 The final step is to divide by 3 leaving x = 5 Copy into your notes Rule for Solving x on both sides 6x = 2x + 10 Remove x from one side In the above example -2x from both sides Clean up by having x on one side only (make 2 step equation): collect x on side with most x’s Step 1) Opposite on both sides (furthest away first) Step 2) Cancel one side Step 3) Solve other side Step 4) Repeat until left with x on its own Check your answer Your Turn: solve the following 1) 4x = x + 9 2) 6x = 2x + 20 3) 7x = 3x + 8 4) 3x = x – 10 5) -2x = 4x - 60 x=3 x=5 x=2 x = -5 x = 10 Equations with x and numbers on both sides 8 xxxxx xx 20 5x + 8 = 2x + 20 Example: Solve 5x + 8 = 2x + 20 Subtract 2x from both sides 3x + 8 = 20 Subtract 8 from both sides 3x = 12 Divide both sides by 3 x=4 Your Turn: Solve the following 1) 2x + 7 = 4x - 13 2) 8x + 3 = 12x – 17 3) 5x – 9 = x + 15 4) 3x + 5 = 33 – x x = 10 x=5 x=6 x=7 Questions to do from the books x and numbers on both sides Achieve Merit Gamma P45 Ex4.03 Q1 - 12 P45 Ex4.03 Q13 - 18 CAT 1.2 P18 Q119 – 122, 131, 132 Excellence SLO Solving equations with brackets Copy into your notes Rule for solving equations with brackets 6(x + 2) = 30 Expand brackets first In the above example this will leave 6x + 12 = 30 Expand brackets Clean up by having x on one side only (make 2 step equation) Step 1) Opposite on both sides (furthest away first) Step 2) Cancel one side Step 3) Solve other side Step 4) Repeat until left with x on its own Check your answer Example 1 multiply out the brackets: subtract 6x from both sides: 3(2x + 4) = 10x 6x + 12 = 10x -6x -6x 12 = 4x divide both sides by 4: ÷4 ÷4 3=x x=3 Can you explain why we subtract 6x not subtract 12? Copy into your notes Example 2 2(3x – 5) = 4x To solve this multiply out the brackets: 6x –10 = 4x +10 add 10 to both sides: +10 6x = 4x + 10 - 4x - 4x subtract 4x from both sides: 2x = 10 ÷2 ÷2 divide both sides by 2: x=5 We add 10 and not subtract 6x as there are more x’s on the left Your Turn Question 6(x – 3) = 12 3(2x – 6) = 30 4(3x – 1) = 8 7(3x – 11) = 70 Answer 5 8 1 7 Questions to do from the books brackets Achieve Gamma P46 Ex 4.04 CAT 1.2 P18 Q123 – 126, 128, 130 Merit Excellence SLO Merit: Solving equations with fractions Copy into your notes Solving equations with fractions 𝑥+3 5 =2 Multiply both sides by denominator(s). In the above example, multiply both sides by 5. This will leave x + 3 = 10 Get rid of fractions by multiplying everything by denominator(s) Expand brackets Clean up by having x on one side only (make 2 step equation) Step 1) Opposite on both sides (furthest away first) Step 2) Cancel one side Step 3) Solve other side Step 4) Repeat until left with x on its own Check your answer Solving equations with fractions 3 x 4 –5=9–x We can remove the 4 from the denominator by multiplying both sides of the equation by 4. 3 4( x – 5) = 4(9 – x) 4 expand the brackets: 3x – 20 = 36 – 4x add 4x to both sides: 7x – 20 = 36 7x = 56 add 20 to both sides: x=8 divide both sides by 7: Example 2x + 7 =x–1 5 To remove the 5 from the denominator we multiply both sides of the equation by 5. 2x + 7 = 5(x – 1) expand the brackets: 2x + 7 = 5x – 5 5x – 5 = 2x + 7 add 5 to both sides: 5x = 2x + 12 subtract 2x from both sides: 3x = 12 divide both sides by 3: x=4 swap sides: Example: Solve 3x + 2 = 11 – x 4 In this case we must start by multiplying both sides of the equation by 4. 3x + 2 = 4(11 – x) Multiply out the brackets: 3x + 2 = 44 – 4x Add 4x to both sides: 7x + 2 = 44 Subtract 2 from both sides: Divide both sides by 7: 7x = 42 x=6 Copy into your notes Example 1: 2 1 x = x + 1 3 2 Get rid of ‘over two’, multiply both sides by 2. 2 1 2( x) =2( x + 3 2 4 𝑥=x+2 3 1) expand the brackets: Get rid of ‘over 3’, multiply both sides by 3. 12 𝑥 3 = 6 x 2 +6 4x = 3x + 6 x=6 Simplify fractions. subtracting 3x from both sides: Copy into your notes Example 1 (Quicker method): 2 1 x = x + 1 3 2 Step 1: Get rid of factions by multiplying both sides by 2 and then by 3. The quick way is to multiply both sides by multiplying by the lowest common multiple of the two denominators. E.g. What is the lowest common multiple of 3 and 2? The lowest common multiple of 3 and 2 is 6. 2 6(3x )= 12 𝑥 3 1 6(2x + 1) 6 = 2x + 6 expand the brackets: subtracting 3x from both sides: 4x = 3x + 6 x=6 Example 2: 5x – 3 2x – 1 = 4 3 Either multiply both sides by 3 and then by 4 or the quick way: multiply by 12 4 3 12(5x – 3) 12(2x – 1) = 41 31 which simplifies to: expand the brackets: 3(5x – 3) = 4(2x – 1) 15x – 9 = 8x – 4 add 9 to both sides: 15x = 8x + 5 7x = 5 5 divide both sides by 7: x= 7 Although this answer could be written as a rounded decimal, it is more exact left as a fraction. subtract 8x from both sides: Copy into your notes We have seen that simplifies to Cross multiplication 5x – 3 2x – 1 = 4 3 3(5x – 3) = 4(2x – 1) How could we perform this simplification in one step? Doing this in one step in often called cross multiplication. Take care: Cross multiplication only works with a fraction either side of the equal sign. You can not use it in the following example: 3𝑥+5 8 4 + 6 = 𝑥 −6 This 4 prevents us using the cross multiplication method. Sometimes the unknowns appear in the denominator. For example, 4 5 = (x + 3) (3x – 5) In this example, we can use cross multiplication to give: 4(3x – 5) = 5(x + 3) expand the brackets: subtract 5x from both sides: add 20 to both sides: divide both sides by 7: 12x – 20 = 5x + 15 7x – 20 = 15 7x = 35 x=5 Press play to show cross multiplication Equivalent equations Your Turn: Solve Question 𝑥+7 3 =5 𝑥+3 3𝑥 + 1 = 2 4 2 𝑥 + 7 = 17 3 3 2 𝑥+7= 𝑥+8 4 3 Answer x=8 x=5 x = 15 x = 12 Questions to do from the books fractions Achieve Merit Gamma P54 Ex4.08/9/10 CAT 1.2 P19 Q133 – 138 Excellence