Simplifying algebraic expressions
- 1. SIMPLIFYING ALGEBRAIC EXPRESSIONS •ADDING LIKE TERMS •MULTIPLYING LIKE TERMS •SIMPLIFYING EXPRESSIONS •REPLACING LETTERS WITH NUMBERS TO GET A VALUE FOR AN EXPRESSION •USING FORMULAS •REMOVING BRACKETS
- 2. ADDING LIKE TERMS 3 + 3 + 3 + 3 Can be written in the shorter form 4 X 3 Meaning that we have 4 bundles of 3’s This gives the VALUE 12 a + a + a + a Can be written in the shorter form 4a We cannot give a value for 4a until we are given a value for a
- 3. If a = 7 Then by replacing a with 7 4a = 4 x(7) = 28 So a value for the expression 4a when a = 7 is 28
- 4. 3a + 2a = a+a+a + a+a = 5a Some examples Simplify the following expressions Combine all the a’s together to get one term 5b – 2b = b + b + b + b + b - b + b = 3b Take 2 b’s away from the 5 b’s
- 5. Simplifying expressions with numbers and different letters Simplify 3a + 2b + 5a + 3b + 7 3a + 5a 2b + 3b+ WE can only add or subtract like terms to each other 8a 5b+ 8a + 5b +7 + 7 + 7
- 6. Multiplying like terms 2 X 2 X 2 X 2 X2 Can be written in the shorter form 25 This means that 2 is multiplied by itself 5 times This gives a value 32 a X a X a X a X a Can be written in the shorter form a5 We cannot get a value for a5 until we are given a value for a
- 7. If a = 7 then a5 = 75 =16807 So a value for the expression a5 when a = 7 is 16807
- 8. Some examples Simplify the following expressions 3a3 x a = 3 x a x a x a x a We now have 3 times a multiplied by itself 4 times = 3 a4 3b x 2b = 3 x 2 x b x b Multiply the numbers together and multiply the letters together= 6 x b2 Write product without multiplication sign = 6 b2
- 9. 3b2 X 4b3 3X4 b2 X b3 X 12 X bXb X bXbXb 12 X b5 b2 X b3 = b2+3 = b5 Multiply the numbers together Add the powers Examples with numbers and letters
- 10. Simplify 4r x 6s x r2 x s3 = 4 x 6 r x r2 =24 r3 s4 When we multiply letters and numbers first put the numbers next the letters in alphabetical order and we do not need the X sign Multiply the numbers together Multiply similar letters together by adding the powers s x s3 24 x r1+2 x x x s1+3 r = r1 REMEMBER
- 11. Order of operations 5 + 2 x 3 = 11 Not 21 In mathematics we have rules Multiplication or division is calculated before addition or subtraction 2 x a + 5 Is simplified to 2a + 5 6 – 2 x b Is simplified to 6-2b
- 12. Have you got it yet ?Simplify the following 1. 2a + 3a 2. 3a + 4b + 2a + 5b 3. b x b x b x b 4. b x b x c x c x c 5. 4b2 x 5b3 6. 3c x 4b 7. 2 c 4 + 3c4 8 . 3m + 2m + 7 9. 6j – 3j + j 10. 3s x 4s x 6 11. 3 x s + 7 12. 6 + 3 x d 13 . 10n – 2 x 3n 14. 8b – 3 x b2 . 5a 5a + 9b b4 b2 c3 20b5 12bc 5c4 5m+ 7 4j 72b2 3s+7 6+3d 4n 8b – 3 b2
- 13. Dividing terms Remember, in algebra we do not usually use the division sign, ÷. Instead we write the number or term we are dividing by underneath like a fraction. For example, (a + b) ÷ c is written as a + b c
- 14. Like a fraction, we can often simplify expressions by cancelling. For example, n3 ÷ n2 = n3 n2 = n × n × n n × n 1 1 1 1 = n 6p2 ÷ 3p = 6p2 3p = 6 × p × p 3 × p 2 1 1 1 = 2p Dividing terms
- 15. Write Algebraic Expressions for These Word Phrases Ten more than a number A number decrease by 5 6 less than a number A number increased by 8 The sum of a number & 9 4 more than a number n + 10 w - 5 x - 6 n + 8 n + 9 y + 4
- 16. Write Algebraic Expressions for These Word Phrases A number s plus 2 A number decrease by 1 31 less than a number A number b increased by 7 The sum of a number & 6 9 more than a number s + 2 k - 1 x - 31 b + 7 n + 6 z + 9
- 17. Finding values for expressions Find a value for the expression 3b when b =7 3b = 3 x (7) = 21 Take out b and replace it with 7 Put 7 inside a bracket 21 is the value of the expression
- 18. Evaluate each algebraic expression when x = 10 x + 8 x + 49 x + x x - x x - 7 42 - x 18 59 20 0 3 32
- 19. Find a value for the expression 3c + b when c = 2 and b = 5 3c + b = 3 x (2) + ( 5) = 6 + 5 = 11 Replace c with 2 and b with 5 11 is the value of the expression Extended activity
- 20. Find a value for the expressions a) pq b) 2p + q c) 5pq +2p d) q2 - p2 when e) 3 ( p + 2q ) f) ( p + q ) 2 when p = 3 and q = 7 a) pq = (3) x (7) = 21 b) 2p + q = 2 x ( 3) + ( 7 ) = 6 + 7 = 13 c) 3pq + 2p = 3 x ( 3 ) x ( 7 ) + 2 x ( 3 ) = 63 + 6 = 69 d) q2 - p2 = (7)2 - (3)2 = 49 - 9 = 40 e) 3 ( p + 2q ) = 3 ( (7) + 2x(3) ) = 3( 7 + 6) = 3 x 13 = 39 f) ( q - p ) 2 = ( (7) – (3) ) 2 = ( 4 ) 2 = 16
Editor's Notes
- #14: Point out that we do not need to write the brackets when we write a + b all over c. Since both letters are above the dividing line we know that it is the sum of a and b that is divided by c. The dividing line effectively acts as a bracket.
- #15: In the first example, we can divide both the numerator and the denominator by n. n ÷ n is 1. We can divide the numerator and the denominator by n again to leave n. (n/1 is n). If necessary, demonstrate this by substitution. For example, 3 cubed, 27, divided by 3 squared, 9, is 3. Similarly 5 cubed, 125, divided by 5 squared, 25, is 5. In the second example, we can divide the numerator and the denominator by 3 and then by p to get 2p. Again, demonstrate the truth of this expression by substitution, if necessary. For example, if p was 5 we would have 6 × 5 squared, 6 × 25, which is 150, divided by 3 × 5, 15. 150 divided by 15 is 10, which is 2 times 5. This power of algebra is such that this will work for any number we choose for p. Pupils usually find multiplying easier than dividing. Encourage pupils to check their answers by multiplying (using inverse operations). For example, n × n2 = n3. And 2p × 3p = 6p2