Junior Certificate School Programme Algebra Workbook For Junior Certificate
Published in 2002 by Junior Certificate School Programme Support Service Curriculum Development Unit Captains Road Crumlin Dublin 12 Phone: 01-4535487 Fax: 01-4020435 Email: jcsp@iol.ie Copyright CDVEC Curriculum Development Unit, 2002 The Junior Certificate School Programme Support Service is funded by the In- Career Development Unit, Department of Education and Skills and the European Social Fund. The Junior Certificate School Programme Support Service is a national Programme sponsored by the Department of Education and Skills and the National Council for Curriculum and Assessment. The Junior Certificate School Programme Support Service is managed and coordinated by the CDVEC Curriculum Development Unit, under the auspices of the Professional Development Service for Teachers Written by Mary G. Cullen, Kylemore College, Dublin. Edited by Aideen Cassidy and Jerry McCarthy. Compiled by Yvonne Canning. Production Aengus Carroll Revised by Mary Clare Higgins 2010 Edited by Denise O Flanagan
Tell us what you think This workbook is open to constant revision and addition and we would appreciate your comments on any aspect of the production. You can get your feedback to us in either of the following ways: Make a note of your comments and give it to your school Junior Certificate School Programme co-ordinator. Drop us a note at the following address: Junior Certificate School Programme Support Service, Curriculum Development Unit, Captains Road, Crumlin, Dublin 12. You can also contact us by: Email: jcsp@iol.ie Phone: 014535487 Fax: 014020435
ALGEBRA AND SIMPLE EQUATIONS A: Introduction to Algebra Scruff s Menu Code T C S M F B Item Cup of tea Cup of coffee Sandwich Soft drink Fish & chips Burger & chips Mr. Scruff uses a code for all the items on his menu. When he takes an order he writes it in code to save time. Examples: 2T + 1C + 3F means 2 Teas, 1 Coffee and 3 Fish and chips. 3M + 2B + 1S means 3 Soft drinks, 2 Burgers and 1 Sandwich.
Exercise 1 Write the codes for the following: 4 Teas, 3 Sandwiches and one Fish and chips. 1 Coffee, 2 Soft drinks and 3 Burger and chips. One Tea, one Coffee, 3 Soft drinks. 4 Burgers and chips, 2 Coffees, 2 Burgers and Chips, 4 Coffees. 1 Coffee, 2 Coffees, 3 Coffees. 3 Teas, 2 Sandwiches, 5 Teas, 6 Sandwiches. 3 Burgers and chips, 3 Fish and chips, 5 Burgers and chips 2 Fish and chips Exercise 2 What do the following codes mean? Example 2T + 3C + 6B means: 2 Teas, 3 Coffees and 6 Burgers and chips. (1) 5M + 6S (2) 7F + 5T + 2M (3) 9B + 3T + 4C +2M (4) 4S + 3F + 8M (5) 2F + 3B +5T (6) 1C + 3T +2S (7) 8C + 5T
(8) 4T + 3S + 1F (9) 6F + 3T (10) 4T + 3T (11) 5S + 4S + 2S (12) 6C + 4C + 7C For one cup of tea or one sandwich we only need to write: T or S Exercise 4 Exercise 3 2 cakes and 2 cakes are four cakes. So 2C + 2C 4C 2C + 2C + C 5C Complete the following as shown in part (a) (a) 3T + 2T + 5T 10T (b) 9R + 1R + 13R (c) 4B + 6B + 9B (d) 8K + 3K + 14K (e) C + 5C + 7C (f) 3P + 11P + 15 P (g) 7S + 4S + 10S + S (h) 5E + 7E + 12E +E
We can use this kind of code for other things: Examples: X+X 2X 2X + 3X 5X Exercise 4 (1) X + X 2X (2) 6X + 2X (3) X + X + X (4) 5X + 4X (5) X +X + X +X (6) 3X + 4X (7) X + X + X + X + X (8) 5X + 3X (9) X + X + X + X + X + X 6X (10) 6X + 3X (11) X + 2X (12) 6X + 2X + X 9X (13) 3X + X (14) 7X + 3X + X (15) X + 6X (16) X + 2X + 2X (17) 4X + X (18) 2X + 4X + X (19) 2X + 8X (20) 4X + 3X + X (21) 2X + X (22) 8X + 4X +2X (23) 1X + 3X (24) 8X + 6X +X (25) 4X + 1X (26) 2X + 2X +5X (27) 6X + 1X (28) 2X + 2X + 6X (29) X + 7X (30) 2X + 3X + 2X (31) 2X + 2X (32) 3X + 3X +2X
(33) 2X + 4X 6X (34) 3X + 3X + 3X 9X (35) 5X + 2X (36) 3X + 4X + 2X (37) 3X + 2X (38) X + 3X + 4X (39) 3X + 3X (40) 2X + X + 5X Sometimes ordinary numbers can be mixed in. If there are ordinary numbers on their own we must keep them on their own. Examples: 2X + X + 1 + 3X + 1 5X + 2 (2X + X + 3X )+ (1 + 1) 5X + 2 2X + 3 + 3X + 5 5X + 8 (2X + 3X) + (3 + 5) 5X + 8 Exercise 5 (1) X + X + 1 (2) X + 2X + 1 (3) X + X + 2 (4) 2X + 2X + 1 + 3 (5) X + X + X + 2 3X + 2 (6) 3X + 2X + 1 + 4 5X + 5 (7) X + X + X + 4 (8) X + 5X + 2 + 1
(9) X + X + 1 + X (10) X + 3X + 2 + 3 (11) 2X + X + 2 + 4 (12) 3X + 2X + 2 + 6 (13) 4X + X + 3 + 3 (14) 3X + 3X + 3 + 6 (15) 5X + X + 3 + 2 (16) 3X + 4X + 2 + 7 (17) X + 5X + 4 + 3 (18) 5X + 3X + 1 + 8 (19) 6X + X + 3 + 4 (20) 1 + 5X + 2X + 2 (21) X + 7X + 5 + 1 8X + 6 (22) 2 + 5X + 3 + 2X 7X + 5 (23) 2X + 2X + 4 + 1 (24) 1 + 7X + 1 + 5X (25) 2X + 3X + 4 + 2 (26) 1 + 6X + 5 + 3X (27) 2X + 4X + 5 + 2 (28) 2 + 7X + 1 + 4X (29) 5X + 2X + 5 + 3 (30) 2 + 6X + 7 + X (31) 3 + 2X + 7 + 3X (32) 4 + 4X + 5 + 3x (33) 3 + 2X + 2 + 4X (34) 4 + 5X + 7 + 2X (35) 3 + 3X + 3 + 2X (36) 4 + 6X + 2 + X (37) 3 + 4X + 4 + 3X (38) 5 + 5X + 2 + 3X
(39) 2 + 8X + 2 + 3X 11X+ 4 (40) 5 + 4X + 4 + 7X (41) 8 + 4X + 2 + 4X (42) 6 + 6X + 3 + 2X (43) 7 + 4X + 3 + X (44) 7 + 2X + 2 + 2X 4X + 9 (45) 7 + 2X + 4 + 4X (46) 7 + 5X + 3 + 2X (47) 9 + 2X + 3 + 5X (48) 7 + 6X + 5 + 5X (49) 9 + 2X + 5 + 4X (50) 7 + 7X + 4 + 3X Exercise 6 2 cakes and 2 sandwiches and 3 cakes and 4 sandwiches 5 cakes and 6 sandwiches 2C + 2S + 3C + 4S (2C + 3C) + (2S + 4S) 5C + 6S Write the answers to these: (i) 2B + 3T + 4T + 5B 7B + 7T (j) 2K + 3P + 4K + 7P (k) 3S + 2E + 4E + 9S (l) 3B + 2T + 4S + 5B + 3T + 3S (n) 2B + 4T + 3B + 4T + 3S + 5S (o) 4T + 3S + 5R + 7T + 8S + 10R
Let s go back to Mr Scruff s Menu to see what happens when we know the amounts the letters stand for: Scruff s Menu Code Item Menu T Cup of tea 5 cents C Cup of coffee 6 cents S Sandwich 7 cents M Soft drink 4 cents F Fish & chips 8 cents B Burger & chips 9 cents It s a very cheap café! Examples: How much does the following order cost? 2T + 2S Answer: 2(5) + 2(7) 10 + 14 24 cents How much will 2M + 3B cost? Answer: 2M + 3B 2(4) + 3(9) 8 + 27 35 cents
Exercise 7 Using the menu on the last page find the cost of the following: Ans 2T + C Ans 3S + 2F Ans 5M + 2T Ans 2B + 7C Ans 4F + 3M Ans 3T + 2B + C Ans 2S + F + B Ans T + 3C + S Ans M + 2F + B Ans S + C + M + 2T
The numbers and letters don t have to be from a menu. Examples: If X 2 and Y 3 work out the following: 3X + 7Y Ans: 3(2) + 7(3) 6 + 21 27 2X + 6Y 4X +Y 5X + 4 Ans: 2(2) + 6(3) 4 + 18 22 Ans: 4(2) + 3 8 + 3 11 Ans: 5(2) + 4 10 + 4 14 Exercise 8 If X 3 and Y 5 work out the following: (a) X + Y (b) 3Y (c) 4X + Y (d) X + 2Y (e) X + X (f) 3X +2Y
(g) X + 3Y (h) Y + 4X (i) 2X + 2Y (j) 6X + 3Y (k) 3X + 4Y + 2 (l) X + 2Y + 5 (m) 6X + 2Y + 1 (n) X + Y + 6 (o) 2X + 3 (p) 2X + Y + 3 (q) 5 + 4X + 3Y (r) 5X + 6 (s) 6X - 3Y (t) 4Y 2X (u) 5X Y (v) 20 4X (w) 15 3Y (x) 25 5Y
TEST 1 If X stands for 3 and Y stands for 4, work out the value of the following expressions to find the answers to the following (a) 2Y (b) 2Y + 3 (c) 3X + 10 (d) 3Y + 11 (e) X + Y (f) 2X + Y (g) X + 2Y (h) 2X + 3Y (i) X + 3Y (J) 3X + Y (k) 5X + 4Y (l) 4X + 5Y (m) 3X + 7Y (n) 10X + 3Y (p) 6X + 7Y
Find the value of the following if A 4 and B 1: (a) A + B (b) 2A (c) 2A + B (d) A + 2B (e) 2B (f) 2A + 2B (g) B + 3A (h) 2B + 3A (i) 2B + 4A (j) B + A (k) 3A (l) 2B + A (m) 7A + B (n) 4B (o) A + 13B (p) 5B + A
Test 2 If X 3 and Y 5 work out the value of these expressions: (a) X + Y (b) 3Y (c) 4X + Y (d) X + 2Y (e) X + X (f) 3X + 2Y (g) X + 3Y (h) Y + 4X (i) 6X + 3Y (j) 9X + 11Y (k) 8X + 3Y (l) 5X+5 If a 2, b 1 and c 3 work out the value of these expressions: (a) 2a (b)3b (c)4c (d) a+c (e)2a+3b (f)a+2c (g)a+b+c (h)a+3b+c (i)a+4b+c (j) 2a+2b+2c (k)3a+b+2c (l) 3a+2b+4c
Simple Equations Sometimes we don t know what the letter stands for but we can work it out. Examples: In your head write it down If X + 4 10 Ans: X 6 because 6 + 4 10 If X + 5 12 Ans: X 7 because 7 + 5 12 Step by step work it out X + 4 10 X + 4 4 10 4 X 6 X + 5 12 X + 5 5 12 5 X 7 If X + 4 5 Ans: X 1 because 1 + 4 5 X + 4 5 X + 4 4 5 4 X 1 f X + 9 14 Ans: X 5 because 5 + 9 14 X + 9 14 X + 9 9 14 9 X 5 These are known as equations because we have to find the value of X by making both sides equal
Exercise 9 Find out what X is in each of the following. Treat them like a Think of a number puzzle. In your head - write it down or Step by step work it out (1) X + 1 9 (2) X + 7 12 (3) X + 2 4 (4) X + 5 30 (5) X + 3 7 (6) X + 21 27 (7) X + 3 21 (8) X + 8 32 (9) X + 4 5
(10) X + 6 36 (11) X + 8 15 (12) X + 6 23 Change the sign What if there is a - (minus) sign between the X and the number? Examples: In your head write it down X 4 6 Ans: X 10 because 10 4 6 or X 6 + 4 10 X 5 3 Ans: X 8 because 8 5 3 or X 3 + 5 8 X 2 7 Ans: X 7 + 2 9 Step by step work it out X 4 6 X 4 + 4 6 + 4 X 10 X 5 3 X 5 + 5 3 + 5 X 8 X 2 7 X 2 + 2 7 + 2 X 9
Exercise 10 Find the value of X in each of the following: (1) X 1 4 (2) X 10 18 (3) X 2 6 (4) X 11 16 (5) X 3 8 (6) X 13 14 (7) X 4 10 (8) X 15 12 (9) X 5 12 (10) X 17 10 (11) X 7 14 (12) X 19 11
When there is a number in front of X (Example: 2X, 3X, 9X) Examples: Find the value of X In your head write it down Step by step work it out 2X 8 X 4 because 2 x 4 8 9X 72 3X 18 X 6 because 3 x 6 18 9X 72 X 8 because 9 x 8 72 1X 8 or X 8
Exercise 11 Work out the value of X in each of the following: You may use In your head write it down or Step by step work it out but you must show your work 1. 4X 24 2. 5X 30 3. 7X 42 4. 6X 36 5. 3X 21 6. 7X 21 7. 9X 54 8. 10X 50 9. 12X 108 10. 8x 88 11. 15X 75 12. 17X 51
When you put together what you already know about simple equations you should be able to solve equations like 2X + 3 13 Example 1: Example 2: 2X + 3 13 so 2X + 3 3 13 3 so 2X 10 3X + 4 16 so 3X + 4 4 16 4 so 3X 12 so..x 5 so.x 4 Exercise 12 Find out what X is in each of the following: (a) 3X + 1 7 (n) 7X + 4 39 (b) 2X + 5 11 (o) 8X + 19 75 (c) 4X + 2 10 (p) 10X + 9 79 (d) 3X + 8 17 (q) 2X + 2 28
(e) 5X + 4 34 (r) 3X + 3 63 (f) 7X + 3 31 (s) 4X + 5 121 (g) 3X + 2 14 (t) 14X + 2 30 (h) 4X + 3 15 (u) 11X + 6 94 (i) 5X + 8 28 (v) 4X + 16 36 (j) 3X + 6 21 (w) 4X + 14 70 (k) 2X + 9 19 (x) 3X + 2 26
(l) 8X + 7 47 (y) 5X + 16 46 When the sign in front of the number is minus (-) Example 1: Example 2: 2X - 3 13 2X - 3 + 3 13 + 3 2X 16 3X 4 11 3X 4 + 4 11 + 4 3X 15 X 8 X 5 Exercise 13 Find the value of X in each of the following: (1) 3X 1 8 (2) 7X 4 38 (3) 2X 5 9 (4) 8X - 19 21 (5) 4X 2 10 (6) 10X 9 41 (7) 3X 8 3 (8) 2X 2 20
(9) 5 X 4 31 (10) 3X 3 33 (11) 7X 3 25 (12) 4X 5 31 (13) 3X 2 16 (14) 14X 2 26 (15) 4X 3 17 (16) 11X 6 49 (17) 5X 8 7 (18) 4X 16 4 (19) 3X 6 21 (20) 4X 14 10
(21) 2X 9 1 (22) 3X 2 25 (23) 8X 7 33 (24) 5X 16 34 (25) 9X 3 15 (26) 9X 18 63
Square Numbers Sometimes we see a small two above a number or letter: Examples: 4 2 X 2 Y 2 3 2 This means we are being asked to square the number or letter. Let s take the number 4 2 and make a picture of it. When we make a square out of the number 4, we end up with 16 small boxes. This is the same as 4 2 4 x 4 16 So to square a number means multiply the number by itself so 3 2 3 x 3 9 and X 2 X x X X 2
If we are asked to square a letter we can only get the answer if we are told what the letter stands for. Example: (Remember X 2 X x X) or x multiplied by x Find the value of X 2 when X 3 If X 3, then X 2 3 x 3 9 Find the value of X 2 when X 4 If X 4 then X 2 4 x 4 16 Find the value of X 2 when X 10 If X 10 then X 2 10 x 10 100 Find the value of X 2 when X 20 If X 20 then X 2 20 x 20 400 Find the value of X 2 when X 30 If X 30 then X 2 30 x 30 900 Exercise 14 Find the value of X 2 when X 9 Find the value of X 2 when X 11 Find the value of X 2 when X 2 Find the value of X 2 when X 12 Find the value of X 2 when X 5 Find the value of X 2 when X 13
Find the value of X 2 when X 6 Find the value of X 2 when X 14 Find the value of X 2 when X 7 Find the value of X 2 when X 15 Find the value of X 2 when X 8 Find the value of X 2 when X 16 Sometimes we can mix squared letters with ordinary letters and numbers. Examples: Find the value of X 2 + 3X + 2 when X 2 Method 1: Write each part under each other Fill in the value for x Work out each part separately and add X 2 2 x 2 4 + 3X + 3 x 2 + 6 Method 2: Put a bracket around each part Fill in the value for x Work out each part and add X 2 + 3X + 2 (2 x 2) + (3 x 2) + 2 (4) + (6) + 2 12 + 2 + 2 + 2 So X 2 + 3X + 2 12 Exercise 15 Now try these yourselves: (a) Find the value of X 2 + X + 4 (h) Find the value of X 2 2X 5 when X 2 when X 4
(b) Find the value of X 2 + 3X + 4 (i) Find the value of X 2 2X + 4 when X 6 when X 5 (c) Find the value of X 2 + 2X 2 (j) Find the value of X 2 2X 6 when X 4 when X 4 (d) Find the value of X 2 3X + 4 (k) Find the value of X 2 + X + 3 when X 5 when X 2 (e) Find the value of X 2 + 6X + 10 (l) Find the value of X 2 3X + 9 when X 6 when X 3 (f) Find the value of X 2 3X + 2 (m) Find the value of X 2 + 5X 24 when X 4 when X 3 (g) Find the value of X 2 2X + 4 (n) Find the value of X 2 4X 12 when X 6 when X 6