S1 Revision/Notes Algebra Unit 2 Solving Equations Algebra History While the word algebra comes from the Arabic language (al-jabr, ال ج بر literally, restoration) and much of its methods from Arabic/Islamic mathematics, its roots can be traced to earlier traditions, most notably ancient Indian mathematics, which had a direct influence on Muhammad ibn Mūsā al-khwārizmī (c. 780-850). He learned Indian mathematics and introduced it to the Muslim world through his famous arithmetic text, Book on Addition and Subtraction after the Method of the Indians. He later wrote The Compendious Book on Calculation by Completion and Balancing, which established algebra as a mathematical discipline that is independent of geometry and arithmetic. Key Vocabulary: Constants A fixed quantity that does not change. For example: 3, 6, 2.5, Variables A variable is a symbol that we assign to an unknown value. It is usually represented by letters such as x, y, or t. For example, we might say that l stands for the length of a rectangle and w stands for the width of the rectangle. We use variables when we need to indicate how objects are related even though we may not know the exact values of the objects. For example, if we want to say that the length of a rectangle is 3 times the length of its width then we can write l = 3 w Coefficients The coefficient of a variable is the number that is placed in front of a variable. For example, 3 w can be written as 3w and 3 is the coefficient. Coefficient
Terms A term can be any of the following: a constant: e.g. 3, 10, the product of a number (coefficient) and a variable: e.g. 3x, 11y, the product of two or more variables: e.g. x 2, xy, 2y 2, 7xy Like terms are terms that differ only in their numerical coefficients. For example: 3a, 22a, - 15a are like terms. Expressions An expression is made up of one or more terms. For example: 3w + 4xy + 5 Solving Equations During primary school you will probably have visited equations for the first time using the cover up method i.e. 3 + * = 9 when you would say to yourself 3 added to what gives you 9. The answer would be 6. Or 4 x * = 20 you would say to yourself 4 multiplied by what gives you 20. The answer would be 5. Instead of using * to represent a missing value, mathematicians tend to use letters instead. is the most popular letter used by Mathematicians. There are various ways of solving equations. In Maths we use inverse operations to solve equations as will be outlined through examples below. We need to know the opposite of adding is subtracting and vice versa and the opposite of multiplication is division and vice versa. At this stage we do not need to worry about negative numbers, fractions or brackets, although extension questions are available should you want to try. Solving equations is different at each level and we should attempt the questions which we feel confident doing. Examples with working at each level are given below:
2 nd level (1 line of working) Solve (a) (b) (c) - 8-8 + 2 + 2 3 3 3 rd level (2 lines of working) Solve (a) (b) - 8-8 +2 +2 3 3 4 4 4th level (3 lines of working) Solve (a) (b) - 2x - 2x - x - x - 4-4 +3 +3 3 3 6 6
Please attempt the questions below at the appropriate level: 2nd Level Questions Solve the following equations to find x: 1. Solve : (a) x + 3 = 5 (b) x + 5 = 9 (c) x + 9 = 12 (d) x + 2 = 7 (e) a + 2 = 4 (f) y + 3 = 8 (g) p + 7 = 11 (h) c + 4 = 5 (i) b + 7 = 9 (j) q + 8 = 8 (k) d + 5 = 10 (l) x + 1 = 6 (m) c + 4 = 6 (n) p + 6 = 13 (o) a + 2 = 15 (p) y + 5 = 14 2. Solve: (a) 2x = 6 (b) 5x = 20 (c) 8x = 16 (d) 3x = 27 (e) 4a = 16 (f) 7y = 28 (g) 6p = 18 (h) 5c = 25 (i) 9b = 36 (j) 2q = 18 (k) 7d = 70 (l) 4x = 32 (m) 8c = 56 (n) 3p = 15 (o) 5a = 35 (p) 6y = 42 3. Solve : (a) x 3 = 4 (b) x 5 = 1 (c) x 9 = 2 (d) x 2 = 7 (b) a 2 = 4 (f) y 3 = 8 (g) p 7 = 11 (h) c 4 = 5 (j) b 7 = 9 (j) q 8 = 8 (k) d 5 = 10 (l) x 1 = 6 (m) c 4 = 6 (n) p 6 = 14 (o) a 2 = 15 (p) y 5 = 14 4. Jack and Jill compare how much money each has. Jack has 53 and together they have 72. Write an equation using x to show this and solve it to find out how much money Jill has.
5. 9 pizzas cost 45. Write an equation using x to show this and solve it to find out how much a pizza costs. 3rd Level Questions Solve the following equations to find x: (Show all working) 1. Solve : (a) 2x + 3 = 5 (b) 4x + 5 = 9 (c) 3x + 3 = 12 (d) 5x + 2 = 7 (e) 2a + 2 = 14 (f) 5y + 3 = 18 (b) 2p + 7 = 21 (h) 3c + 4 = 16 (i) 6b + 7 = 49 (j) 8q + 8 = 8 (k) 2d + 5 = 35 (l) 3x + 5 = 26 (m) 8c + 4 = 36 (n) 7p + 6 = 55 (o) 12a + 2 = 26 (p) 9y + 5 = 50 2. Solve : (a) 3x 2 = 7 (b) 4x 5 = 11 (c) 2x 9 = 3 (d) 3x 7 = 5 (e) 7a 2 = 12 (f) 5y 3 = 22 (g) 6p 7 = 29 (h) 4c 3 = 29 (i) 8b 7 = 57 (j) 10q 8 = 72 (k) 3d 5 = 31 (l) 9x 1 = 80 (m) 4c 9 = 15 (n) 6p 2 = 40 (o) 5a 2 = 73 (p) 3y 14 = 40 3. Louise has 3 full packets of rolos. She also has 4 spare rolos. When she counts all of her sweets she finds she has 31. Write an equation using x to show this and solve it to find out how many rolos are in each pack. 4. A football coach charges a call out fee of 20 and 30 per hour for his coaching. He earns 140. Write an equation using x to show this and solve it to find out how many hours the coach works for.
4th Level Questions Solve the following equations to find x: (Show all working) 1. Solv : (a) 6y + 3 = y + 18 (b) 5a + 7 = a + 15 (c) 9c + 5 = c + 21 (d) 10x + 1 = 4x + 19 (e) 5b + 3 = 2b + 9 (f) 7n + 6 = 3n + 18 (g) 3x + 2 = x + 14 (h) 9c + 58 = 6c + 73 (i) 16 + 7y = 2y + 31 (j) 15a + 4 = 3a + 76 2. Solve: (a) 6y 3 = 3y + 15 (b) 5a 9 = a + 15 (c) 9c 8 = 4c + 12 (d) 10x 1 = 4x + 5 (e) 5b 3 = 2b + 9 (f) 3n 10 = n + 2 (g) 7x 14 = 3x + 2 (h) 6c 13 = 3c + 59 (i) 7y 16 = 2y + 34 (j) 15a 8 = 3a + 76 3. Tommy is a joiner and he bought 6 boxes of screws, though he already had 40 loose ones. Alf, his apprentice, had 4 boxes of screws as well as 190 loose screws. They found each had the same number of screws. Write an equation using x to show this and solve it to find out how many screws are in a full box.