M7 Class 20 notes Algebra (Basic Equations) 1. Addition and/or subtraction in equations Equations are something that you will constantly be using throughout your math career. Learning and understanding the basic is an integral part of "getting off on the right foot" when dealing with math. This section will help you better understand, work with, and solve equations when they have addition and/or subtraction in them. • • • Changing the order of the addends (numbers you're adding) doesn't change their sum (what they equal when added together). Example: a + (b + c) = (a + b) + c Any number plus 0 (zero) equals itself. Example: a+0 =a If two sides of an equation are equal, you can add or subtract the same amount to both sides, and they will still be equal. Example: a=b a+c=b+c a-c=b-c When solving equations, remember that addition and subtraction are inverse operations - they undo each other (i.e., 10 + 9 - 9 = 10). To solve equations using addition and subtraction, first decide which operation has been applied, then use the inverse operation to undo this (remember to add or subtract from both sides of the equation). Example: 1. Solve: x + 79 = 194 Solution: x + 79 = 194 x + 79 - 79 = 194 - 79 x = 115 You need to get the variable by itself (isolate the variable). 2. Solve: x - 56 = 604 Solution: x - 56 = 604 x - 56 + 56 = 604 + 56 x = 660 You need to isolate the variable. To undo adding 79, subtract 79 from both sides. To undo subtracting 56, add 56 to both sides. 1 M7 Class 20 notes 2. Multiplication and/or division in equations This section will help you understand, work with, and solve equations of a slightly more complex nature - equations involving the use of multiplication and/or division. Order of operations: • • • • The operations inside parentheses () and brackets [] are done first. Then any operations involving exponents Then do all multiplying and dividing from left to right. Finally, do all addition and subtraction from left to right. Multiplication can be written three different ways: 9 * x; 9x; 9(x) A fraction bar is also a division symbol. Changing the order of multipliers (numbers you're multiplying together) doesn't change their product (total when the numbers are multiplied together). Example: ab = ba. Zero times any number is zero and 1 times any number is the number. Examples: x(0) = 0; (0)x = 0; x(1)= x; 1 * x = x. If two sides of an equation are equal, you can multiply or divide each side by the same quantity (number or equation) and it will still be equal. Examples: a = b, c <> 0; ac = bc; (a / c) = (b / c). When solving equations, remember that multiplication and division are inverse operations, therefore they undo each other (i.e., (4 * 8)/8 = 4). To solve equations using multiplication or division, first decide which operation has been applied, then use the inverse operation to undo this (remember to multiply or divide on both sides of the equation). Examples: 1. Solve: 6x = 36 Solution: 6x = 36 (6x) / 6 = 36 / 6 x=6 2. Solve: x / 5 = 10 Solution: x / 5 = 10 5(x / 5) = 10(5) x = 50 You need to get the variable by itself (isolate the variable). To undo multiplying by 6, divide by 6 on both sides. You need to get the variable by itself (isolate the variable). To undo multiplying by 6, divide by 6 on both sides. 2 M7 Class 20 notes 3. Combinations of the basic operations in equations This section will help you understand, work with, and solve complex equations that involve different combinations of multiplication, division, addition, and subtraction. Order of operations: • • • • The operations inside parentheses () and brackets [] are done first. Then any operations involving exponents. Then do all multiplying and dividing from left to right. Finally, do all addition and subtraction from left to right. Multiplication can be written three different ways: 7 * x; 7x; 7(x). A fraction bar is also a division symbol. Also, be sure to refer to the above sections if you have forgotten or need to review any of the other material covered. When solving complex equations, like the ones used in the examples below, be sure to remember that multiplication and division are inverse operations along with addition and subtraction. Therefore, they undo each other (i.e., (5 * 2)/2 = 5 or 10 + 4 - 4 = 10). To solve these equations, first decide which operation has been applied and then use the inverse operation to undo this (remember to apply the operation to both sides of the equation). Example: 1. Solve: 7x - 7 = 42 Solution: 7x - 7 = 42 7x - 7 + 7 = 42 + 7 7x = 49 (7x) / 7 = 49 / 7 x=7 The variable needs to be isolated. To undo subtracting 7, add 7 to both sides. 2. Solve: 5(x + 2) = 25 Solution: 5(x + 2) = 25 [5(x + 2]/5 = 25/5 x+2=5 x + 2 -2 = 5 -2 x=3 The variable needs to be isolated. To undo multiplying by 5, divide by 5 on both sides. Adding 7 hasn't isolated the variable, so we need to continue. To undo multiplying by 7, divide both sides by 7. Dividing by 5 hasn't isolated the variable, so we need to continue. To undo adding 2, subtract 2, from both sides. 3 M7 Class 20 notes Questions in class (ALL questions must be solved using ALGEBRA) (After the class, please do the following questions again) 1. A hockey team has won three times as many games as it has lost. If it has won 84, how many has it lost? 2. You are given three numbers. The second is 5 more than the first and the third is double the sum of the first two. If the sum of these numbers is 75, find the second number. 3. Peter and Mary have a total of 1056 cookies. Peter has three times as many as Mary. How many cookies does Mary have? 4. Anne is two years older than Brigitte and Brigitte is 6 years older than Carol. What is the sum of their ages when Anne is twice as old as Carol? 5. A student receives grades on four examinations of 75, 82, 71 and 84. What grade does the student need on the fifth examination to raise their average to 80? 6. Paul is 5 years younger than Roman and Jules is 10 years younger than the sum of Paul's and Roman's ages. How old is Roman if the three ages add up to 80 years? 7. A, B and C are three numbers. If C - 2A = 50 and B + 3A = 10, then what is the average of the three numbers? 8. Twice the product of two numbers is 108 while one third of their sum is 5. What is the value of the smaller of these numbers? 9. The sum of my age and my sister’s age is 11. The product of our ages is 24. What is the difference of our ages? 10. For what number would subtracting 3 from it and then dividing that result by 9 give the same outcome as first subtracting 9 from it and then dividing the result by 3? 4