Algebra Skills Year 10

Note 1: Expressions An algebraic expression is a statement using symbols. Expressions need to be written as simply as possible. There are rules that should be followed: A multiplication sign is not used eg. r x s = rs The number part is written first in an expression eg. y x 5 = 5y Letters are written in alphabetical order when multiplying eg. 5a x g x d = 5adg Divisions are written as fractions eg. 3p ÷ s =

Note 1: Expressions S = (n – 2) × 180° Algebra is used to express rules using symbols. A letter is used to stand for a number. The letter is placed into a formula to explain what happens. e.g. Write the following as a mathematical expression: The sum of all the angles in a polygon is calculated by subtracting two from the number of sides and multiplying this number by 180° S = (n – 2) × 180° BETA: Ex 7.01 pg 208 Ex 7.02 pg 209 - 212

Starter: simplify the following expressions

Note 2: Substitution = 4 = 7x – 1 when x = 2 = 7  2 – 1 = 14 – 1 = 13 Substitution means replacing a symbol with a value. Remember to follow the rules of BEDMAS. e.g. Calculate the value of these expressions: 7x – 1 when x = 2 = 7  2 – 1 = 14 – 1 = 13 when f = 2 and g = 6 = 4 =

Note 2: Substitution 5x2 - 3x + 2 when x = -3 = 5x(-3)2 - 3  -3 + 2 = 45 - −9 + 2 = 56 BETA: Ex 7.03 pg 215 Ex 7.04 pg 216 Ex 7.05 pg 218

Note 3: Simplifying Multiplication When multiplying terms all numbers and variables can be combined.   e.g. Simplify: 3a x 4b =12ab -5c x 6d x -2e = 60cde BETA: Ex 8.01.pg 231 Homework book: Ex C pg 75

Note 4: Adding and Subtracting Like Terms Like terms can be added or subtracted, if they are the same term and of the same power The sign (+ or -) belongs to the number or variable after it. e.g. Simplify 3a + a – 2a = 4b – 7b + 2b = 7c + 5d – 9c + 2d = 5ef + 6fg – 8fe + 7hg = 2a -b -2c + 7d -3ef + 6fg + 7gh  

Note 4: Adding and Subtracting Like Terms When we add and subtract like terms with powers, we do not change the powers.   e.g. Simplify: k3 – k2 + 3k + 4k3 – 6k2 = 5k3 – 7k2 + 3k BETA: Ex 8.02 pg 232 Ex 8.03 pg 233 Homework book: Ex B pg 73

an Note 5: Powers e.g. 22 x 23 = (2 x 2) x (2 x 2 x 2) = 25 (22+3) y7 exponent, power base y xn means use ‘x’ as a factor n times. e.g. p x p x p = p³ Multiplication When multiplying numbers with the same base, add the powers. e.g. 22 x 23 = (2 x 2) x (2 x 2 x 2) = 25 (22+3) y3 x y4 = y7 BETA: Ex 8.05 pg 242 Ex 8.06 pg 245 Ex 8.07 pg 246 3q x 7q = 21q2 6r3 x 5r2s = 30r5s

Note 6: Square Roots = 6 = = x3 =  x42  y102 = 8x2y5 = 8x7y4 When simplifying the square root of an algebraic expression take the square root of the number divide the power of each variable by 2.   Examples: Simplify: = = 6 = = x3 =  x42  y102 = 8x2y5 =  x142  y82 BETA: Ex 8.08 pg 248 = 8x7y4

Note 7: Expanding Brackets To expand brackets: Multiply the outside term by everything inside the brackets Simplify where possible   e.g. Expand and Simplify: a.) 4(x + 2) b.) −6(3x – 1) c.) x(2x – 3) BETA: Ex 9.01 pg 253 Ex 9.02 pg 254 Ex 9.03 pg 255 = 4x + 8 d.) 5x – 2(6 +3x) = 5x – 12 −6x = −18x + 6 = −x – 12 e.) 4(2x – 7) −2(6 −x) = 2x2 − 3x = 8x – 28 −12 +2x = 10x − 40

Note 8: Factorising d.) 6x + 21 = 4(a+b) = 3(2x +7) = 3(p – q +r) BETA: Pg 257 onwards Ex 9.04, 9.05, 9.06, 9.07 Factorising is the opposite of expanding – putting brackets back into the algebraic expression: Look for the highest common factor in the numbers and place it outside the brackets. Look for any variables (letters) that are common. Take the lowest power and place it outside the brackets.   e.g. Factorise: a.) 4a + 4b b.) 3p – 3q + 3r c.) 4x + 8y + 12x d.) 6x + 21 = 4(a+b) = 3(2x +7) = 3(p – q +r) e.) 24x - 32 = 4(x + 2y + 3z) = 8(3x – 4)