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... 11 Simultaneous Equations... 1 Elimination method... 1 Rearranging equations... 15 Sequences... 16
Collection of like terms When collecting like terms remember we can collect together equivalent letters i.e. 7 a + a = 10a and we can collect together numerical terms i.e. 4 + 8 = 9 However we cannot collect together terms that are not alike i.e. 4 a + 5b cannot be simplified. Nor can a 4 Simplify each of the following 9 a + b 4a + 5b = 5a + 8b a) {Remember to take note of the sign in front of each b) c) d) letter} 6x y + x 6y = 7x 8y d + 4 e f + e 7d + 4 f = 6d + 7e + f p 8q 7 p + q = 5p 5q Solving simple equations Solve each of the following equations a) b) c) d) e) f) g) h) x 4 = 17 5 x + = x + 7 7x 5 = 4x 9 1 x = x 19 ( x 1) 1 4 = ( x + 5) 7 = x + 5 = 5 ( x + 7) 5 = 9
Golden rules The equation starts balanced and must remain balanced! So whatever you do to one side you must do to the other side. i.e. (a) x 4 = 17 x 4 + 4 = 17 + 4 {Add 4 to both sides of the equation} x = 1 x = 1 {Simplify} { D i v i d e b o t h s i d e s b y } x = 7 {Simplify giving answer} An alternative way of thinking! When moving a number from one side of the equal sign to the other we perform the opposite operation. The opposite of Addition is subtraction and visa versa. The opposite of Multiplication is Division and visa versa. (b) 5 x + = x + 7 5x + = x + 7 5x = x + 7 5x x = 5 x = 5 x =. 5 {take the over and subtract } {take the x over and subtract} {Simplify} (c) 7x 5 = 4x 9 7x 5 = 4x 9 7x = 4x 9 + 5 7x 4x = 4 x = 4 4 x = = 1. (d) 1 x = x 19
1 x = x 19 1+ 19 x = x 0 = x + x 0 = 5x x = 4 {Move the19 over and add!} {Move the x over and add} {Simplify} (e) ( x 1) 1 4 = 4 ( x 1) = 1 4x 4 = 1 4x = 1 + 4 4x = 16 x = 4 {remove brackets first} {now we can rearrange as before} (f) ( x + 5) 7 = ( x + 5) = 7 6x + 15 = 7 6x = 7 15 6x = 8 8 4 x = = = 1. 6 (g) x + 5 = 5 x + 5 = 5 x + 5 = 15 x = 15 5 x = 10 x = 5 {take the over and times} (h) ( x + 7) 5 = 9
( x + 7) = 9 5 ( x + 7) = 45 x + 1 = 45 x = 45 1 x = 4 x = 8 {Take the 5 over and times} {Remove the bracket} Factorisation Factorise each of the following: a) b) c) d) x y + x pq + 4 p q 4m + 10m 6mn 7a b + a b abc Here we cannot place into two brackets since it does not follow the pattern of followed by x, followed by a constant! However we have a common factor. So place the common factor outside a bracket. a) x Since x xy = x y y + x = x xy ( + ) Since x = x x b) c) d) ( q p) pq + 4 p q = pq + 4 ( + 5m n) 4m + 10m 6mn = m ( 7a + ab c) 7a b + a b abc = ab
Inequalities Solve each of the following inequalities a) b) c) d) e) x < 9 x + 7 19 4x 1 > x 8 5 x + x + 7 ( 4x 1) 9 < f) 5 + x 7 Solving inequalities can be like solving equations. However don t forget to write the correct symbol; and not the equal sign! a) x < 9 x < 9 x < 9 + x < 1 x < 6 {Add to the other side} {Simplify} b) x + 7 19 x + 7 19 x 19 7 x 1 x 4 c) 4x 1 > x 8 4x 1 > x 8 4x > x 8 + 1 4x x > 7 x > 7
d) 5 x + x + 7 5x + x + 7 5x x + 7 5x x 4 x 4 x e) ( 4x 1) 9 < ( 4x 1) < 9 8x < 9 8x < 9 + 8x < 11 x < 11 8 f) 5 + x 7 5 + x 7 5 + x 1 x 1 5 x 16 Whole number Given that n is an integer, find the values of n such that 7 n < 6.5 n < Hence n can equal {divide throughout by },, 1,0,1,
List the values of n such that n is an integer value and a) < n < b) 1 < n 5 c) 4 n < Here we are not asked to find the solution by solving an equation but by listing the possible solutions. An integer value means possible whole number answers whether positive or negative. n n = {, 1,0,1} a) < < means n can be 1, 0 or 1 we write n n = { 0,1,,,4,5 } b) 1 < 5 Answer: n n = { 4,,, 1,0,1, } c) 4 < Answer: Graphs 1. The straight line Draw the graph of y = x 4 for values of from - to For any straight line we can get away with plotting three points and then a line through these three points. Three points are selected to make sure we have only one straight line. X 0 y 1 4 5 y=(-)-4 = -9-4 =-1 y=(0)-4 = -4 y=()-4 =9-4 =5
y 6 4-0 x - -4-6 -8-10 -1-14 Draw the graph of x + y = 7 x for values of from 4 to X 4 0 y 7 4 4+y=7 0+y=7 +y=7 y 10 8 6 4-0 4 6 8 x -
. The quadratic curve The quadratic Curve could appear on both papers but for non-calculator papers it will be a very basic curve. (a) Complete the table of values of y given y = x x 5 x 4 1 0 1 4 y 19 10 5 6 (b) Draw the graph of y = x x 5 (c) Use the graph to state the values of x for which x x 5 = 0 (d) State the minimum value for y on the graph. a) X 4 1 0 1 4 y 19 10 5 6 5 y = ( ) ( ) 5 = 4 + 4 5 = y = ( ) ( ) 5 = 4 4 5 = 5 b) y 0 15 10 5-4 - - -1 0 1 4 x -5 c) the curve equals zero at the points where the curve cuts the x-axis x = 1.4, x =.4
d) Minimum value of curve is where the curves gradient changes from going down to going upwards. i.e. y = 6 when x = 1 a) Complete the table below for the graph of y = x x x 1 0 1 4 y 7 14 0 1 0 b) Draw the graph of y = x x c) Draw on the same axis the graph of y + x = 14 d) Use the graph to obtain a suitable approximation to solution of the simultaneous equations given by y = x x and y + x = 14 a) x 1 0 1 4 y 7 14 5 0 1 9 0 Type in -1 and exe Then type Ans Ans b) Type in and exe Then type Ans Ans y 0 5 0 15 10 5 - - -1 0 1 4 x -5
c) X 0 1 y 14 1 8 y+0=14 y+=14 y+6=14 Line drawn on the graph in part (b) d) The curve y = x x and the line y + x = 14 are equal at the points of intersection. ( ) (.9,8.1) Which occur at.4,19 and approximately. Simultaneous Equations Elimination method Remember If the signs are different we add the equations, and If the signs are the same we subtract. Solve the equations x + x y = 10 y = 6 Step1: Eliminate y by adding the two equations x + x x x y y = 10 = 6 = 16 = 8
Step : Place the value of into the first equation x + 8 + y = 10 y = 10 y = Step : Write down your solution x = 8, y = Solve the equations x + y = 8 x y = 5 Step1: Multiply an equation by a number so that we can eliminate x or y. e.g. multiply the first equation by to give 4 x + y = 16 Step : Eliminate the y by adding the two equations 4x + y = 16 x y = 5 7x = 1 x = Step : Place the value of into the first equation x + 6 + y y y = 8 = 8 = Step4 : Write down your solution x =, y =
Rearranging equations C = πr C = πr...rearrange to make r the subject C = r π divide both sides by π A = πr...rearrange to make r the subject A = r π divide both sides by A = r π square root both sides v = u + at...make a the subject v u t = a take u to the other side divide both sides by t y = x... make the subject y = x y = x y + = x remove the square root by squaring the other side take the over to the other side y + = x finally square root to find x
Sequences i) The simplest sequence is the sequence of natural numbers i.e. 1,,, 4, 5, 6,... ii) Other simple sequences are 1,, 5, 7, 9,... [Odd numbers], 4, 6, 8, 10,... [Even numbers],, 5, 7, 11, 1,...[Prime numbers] iii) Continuing a sequence Find the next two terms in the following sequences: a), 5, 8, 11,... Answer = 14, 17, {add three to the previous term} b) 1, 4, 9, 16,... Answer = 5, 6 {square numbers} c) 1, 1,,, 5, 8,... Answer = 1, 1 {sum of the last two gives the next term} iv) The nth term of a sequence The nth term of any sequence can be written in many forms n th term = U n = T n = etc.. in each case the nth term is a method of calculating directly a given term for a sequence.
Given the nth term of a sequence is the sequence. U n = 4n 1, write down the first four terms to In this case the first term U 1 is found by replacing n with 1 in the formula ( ) 1 = 4 1 U1 = 4 1 = Similarly ( ) 1 = 8 1 7 U = 4 = ( ) 1 = 1 1 11 U = 4 = ( 4) 1 = 16 1 15 U 4 = 4 = Hence first four terms are, 7, 11, 15 Using the nth term U n ( ) = n n +1, write down the first four terms to the sequence. ( + 1) U1 = 11 = ( + 1) 6 U = = ( + 1) 1 U = = ( 4 + 1) 0 U 4 = 4 =, 6, 1, 0 v) Finding the nth term There are an infinite number of expressions for the nth terms, but here are a few pointers to look for. If the sequence goes up by the same amount each time (i.e. its linear) then the U sequence can always be written in the form n = mn + c, where is the common difference and is the number which must be added to obtain the first value.
Find the formula for the nth term of the sequence 5, 8, 11, 14,... Here the common difference is so the formula must involve this would generate the sequence, 6, 9, 1,... which is short each time Obtain an expression for the nth term of the sequence 4, 11, 18, 5,. Here the common difference is 7 so the formula must involve -however -however this would generate the sequence 7, 14, 1, 8,... which is over each time U n U n = n + = 7n U n = n U n = 7n