Mathematics Revision Guide. Algebra. Grade C B

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1 Mathematics Revision Guide Algebra Grade C B 1

2 y 5 x y 4 = y 9 Add powers a 3 a 4.. (1) y 10 y 7 = y 3 (y 5 ) 3 = y 15 Subtract powers Multiply powers x 4 x 9...(1) (q 3 ) 4...(1) Keep numbers without letters separate 4a - 2a 3y -2y 2a - 2y = 6a + y 4a + 3y + 3p + 2q p + 2q (2) 4 x 5 y x p 3n 2p...(1) 4y x 5p = 20yp a 2 x a 1 fully 4 x 5 y 4 x y 3 3p 5 q 4p 3 q 2...(2) 5y 4 a 2 x 4y 3 a = 20y 7 a 3 fully 40 5 y19 y10 40y 19 t 6 = 8y 10 t 4 t 6 t 2 4 3q 2q q (2) 5y 9 t 2 2

3 Expand A) 2(y + 5) = 2y + 10 Algebra Revision Notes Expand - remove the brackets by multiplying 2 x y 2 x 5 Expand 3y(y + 4)...(2) B) 3(x + 10) + 2(3x 2) = 3x x - 4 = 9x + 26 Expand Expand 3x + 6x (2x + 5) + 2(3x 2)...(2) (x+4)(x+5) X +4 X x 2 +4x +5 +5x +20 Draw a Grid X 2 + 4x + 5x + 20 X 2 + 9x + 20 Expand and simplify (x 6)(x + 4) (2) (x+10)(x-2) Draw a Grid Expand and simplify (2x + 5)(3x 2) X +10 X x 2 +10x -2-2x -20 X 2 +10x 2x 20 X 2 +8x 20 (2) 3

4 Factorise put the brackets back in A) 5x + 30 = 5(x + 6) Factorise 3y 12 5 is the highest factor/number that goes into both 5 and (1) B) X 2 5x = x(x -5) X is the common term that appears in both parts of the equation. Factorise x 2 10x....(Total 1 mark) B) 8x xy = 4x(2x + 3y) Factorise completely 5x² + 10xy The highest factor/number that goes into 8 and 12 is 4...(2) X is also a common term that appears in both parts of the equation A) X 2 + 6x + 8 = (x + 4) (x + 2) Factorise x 2 + 7x + 10 SECOND - identify which pair of numbers will add or subtract to give you the middle number. FIRST - Identify all the factors (in pairs) that go into 8. 1, 8 2, 4...(2) A) X = (x + 10)(x -10) Factorise x 2 + 2x 15 1, , 10 25, 4 20, 5 In this example there is no amount of x in the middle of the equation. Therefore you need to identify numbers which you can add together or take away to give the answer 0..(Total 2 marks) 4

5 Solve - find the value of the letter/term A) 2x + 3 = To solve - Work the equation backwards and do the opposite Solve 4x + 3 = becomes 3 x2 becomes 2 X = 7/2 or 3.5 B) 4x + 2 = 2x + 18 Subtract the smaller amount of x from each side subtract 4x + 2 = 2x + 18 This is a balancing equation as it has x on both sides. Solve 5t 4 = 3t + 6 2x + 2 = To solve - Work the equation backwards and do the opposite +3 becomes 3 x2 becomes 2 X = 16/2 = 8 C) 2(5x + 3) = 3x - 22 Expand first 10x + 6 = 3x 22 7x + 6 = -22 Subtract the smaller amount of x from each side Solve 3(x 4) = x = X = -4 5

6 N th Term number patterns Calculate the n th term A) 5, 7, 9, 11, n How do you get from the difference to the first number Find the difference between the numbers and then add a n to it. Calculate the n th term 3, 9, 15, 21, 27 Write the first 5 terms Goes up in 5 s Then add 1 B) 5n + 1 Write the first 5 terms for 7n - 2 (1) (2) (3) (4) (5) (1) (2) (3) (4) (5) Substitution change the letter into a number x = 10 y = -4 x = 17 y= -2 Find the value of Find the value of 4x + 3y 3x + 6y 4 x 10 = 40 3 x -4 = = 28 6

7 Inequalities - state values which satisfy / solve State values that satisfy the inequality A) -1 x < 3 2 x < 3 x is an integer. Include -1 Don t Include 3 Write down all the possible values of x -1, 0, 1, 2... (Total 2 marks) State values that satisfy the inequality B) -4 < x 2 Don t Include 4 Include 2 Solve the inequality 5x < 2x 6-3, -2, -1, 0, 1 C) Solve 4x To solve - Work the equation backwards and do the opposite...(2) becomes 1 x4 becomes 4 Solve the inequality 5x + 12 > 2 X 5 Remember to include the inequality symbol back into your final answer. Use the same one that was in the question. (2) 7

8 Algebraic Graphs straight line Draw a graph for x + y = 4 Draw a table with x values form 2 to 2 X y Start at = = +4 8

9 Algebraic Graphs straight line Draw a graph for y= 2x +1 x + 2 Forming and Solving Equations x + 3 The shape has a perimeter of 53cm. Calculate the value of x to show that this is correct. STEP 1: x x x + 3 +x + 3 4x + 10 STEP 2: Solve the equation Remember that the sides of the shape are the same. Step 1: collect all the terms together Step 2: solve the equation to find out the value of x 4x + 10 = = 40 4 x = 10 9

10 Simultaneous Equations same coefficient When one of the letters has the same coefficient (a) 2x + 3y = 0 (b) x 3y = 9 Add the equations together 3x = 9 Solve the equations x = 9 3 x = 3 Both coefficients of y are the same. +3 and 3. Therefore we can add the equations together to get 0y TIP If the symbols + or - are the same for the term you are trying to eliminate then you subtract the equations from each other. If the symbols are different for each term then you can add the equations together. Now you know the value of x put this back into one of the equations to calculate the value of y. The value of x put back into the first equation (a) (2 x 3) + 3y = y = 0 3y = -6 y = -6 3 y = -2 5a + 3b = 9 2a 3b = 12 a =... b =... (Total 3 marks) 10

11 Simultaneous Equations different coefficient When the coefficients are not the same you have to make them the same! (a) 3x 4y = 11 (b) 5x + 6y = 12 As the coefficients are not the same, we need to make them the same. To do this we can multiply the equation marked (a) by 3 and equation (b) by 2. This will give me 12y for both. (a) x 3 9x - 12y = 33 (b) X 2 10x + 12y = 24 The symbols are not the same, therefore we can add the equations together to get to 0y Add equation (a) to equation (b) 19x = 57 Solve the equation x = x = 3 Put the value of x back into equation (b) to find the value of y I have used equation (b) as it has a + rather than a -. (5 x 3) + 6y = y = 12 Solve the simultaneous equations (a) 2x + 3y = 3 (b) 3x 2y = 28 6y = y = -3 y = -3 6 y = -0.5 x = y = (Total 4 marks) 11

12 Trial and Improvement x 3 + 2x = 110 The solution is between 4 and 5 to 1 decimal place. Use trial and improvement to find a solution. x 3 x = 30 The solution is between 3 and 4. Use a trial and improvement method to find this solution. Give your answer correct to 1 decimal place. x x 3 +2x (4.5) 3 + (2 x 4.5) Too small Too small Too big Less than 110 so 4.7 is closer to 1dp Changing the Subject rearranging the equation Make x the subject y = 2x + t t = ax 5 x x2 +t =y Work the equation forward Make x the subject y -t 2 x Work the equation backwards and do the opposite y - t = x 2 12

Algebra Revision Guide

Algebra Revision Guide Stage 4 S J Cooper 1st Edition Collection of like terms... Solving simple equations... Factorisation... 6 Inequalities... 7 Graphs... 9 1. The straight line... 9. The quadratic curve...