Algebra, Part I. x m x = n xm i x n = x m n = 1

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1 Lesson 7 Algebra, Part I Rules and Definitions Rules Additive property of equality: If a, b, and c represent real numbers, and if a=b, then a + c = b + c. Also, c + a = c + b Multiplicative property of equality: If a, b, and c represent real numbers, and if a=b, a b = c c then ca = cb. Also, ac = bc, and, Product rule for eponents: If is not zero, then m n = m+n Quotient rule for eponents: If m and n are real numbers and is not zero, then m = n m i n = m n = 1 n m Definitions - algebra: a generalization of arithmetic, where letters representing numbers are combined according to the rules of arithmetic, often to solve for an unknown value. - In algebra, you don t have to write the 1:1 =, and 1 =. - 0 = 1: Anything (ecept 0) raised to the power of 0 equals = n : For eample, = 2, = 3, = 4, etc. 1 A bag of rocks from the 1912 eruption of Novarupta volcano. In algebra, we use symbols like to represent the quantity of rocks in the bag. - -n = n. Anything (ecept 0) raised to a negative power equals 1 over anything raised to the opposite of the power. For eample, -2 1 = 2, 1-3 = 3, 1 7 =, etc. 7 - polynomial in one unknown: One term, or a sum of individual terms of the form a n, where a is a real number, is an unknown quantity, and n is a whole number. Eamples include 3 2 (monomial), 2a + 7 (binomial), and z 5 + 3z 2 - z (trinomial). - polynomial in more than one unknown: One term, or a sum of individual terms of the form a n y m z p..., where a is a real number,, y, z, etc. are unknown quantities,

2 and n, m, p, etc. are whole numbers. Eamples include 3 2 y 2 (monomial), 2yz (binomial), and c 2 d h 2 - yz (trinomial). 7A What is Algebra? In the year 830 A.D., an astronomer named Mohammed ibn Musa al-khowarizmi wrote the book Hisab al-jabr w al-muqabalah, meaning science of transposition and cancellation. It was a book about working with mathematical equations. For eample, in the equation 4-3 = 9, applying al-jabr (add +3 to both sides) transposes the equation to 4 = 12. Then, applying al-muqabalah (dividing both sides by 4) cancels the 4 on the left, resulting in = 3. Over time, most of the words were dropped and the word algebra was used. Today, we normally think of algebra simply as arithmetic that uses variables (letters) to represent numbers. Many students struggle with the transition from working with numbers to working with variables. For eample, it s easy to think of a bag of rocks with 20 rocks, but what about a bag with rocks? What does that mean? Well, all it really means is that we don t know the eact amount of rocks in the bag. Or, perhaps the number of rocks varies, because people are constantly taking rocks out and/or putting rocks back in. = a bag of rocks of unknown quantity Thinking of the bag of rocks, if =, then = 2, = 3, etc. We use to mean bag of rocks, 2 means two bags of rocks, 3 means three bags of rocks, etc. The sketch of a bag of rocks is our more concrete way of thinking about unknowns, while using variables like is a more abstract method. A bag of rocks is something real that we can hold, is not. Remember our analogy from Lesson 6? ideas : reality :: abstract : concrete The mathematical idea of helps us consider the reality of a bag of rocks of unknown quantity. Eample 7.1 If = 10, what does equal? solution: There are 3 bags with 10 rocks each, so we multiply to get 3 10 = 30. Eample 7.2 Evaluate 3b if b = 10.

3 solution: Evaluate means find the value of. This problem is actually identical to e. 7.1, it just looks more algebraic. 3b means 3 times b, so replace (substitute) b with 10 and solve to get 3 10 = 30. Eample 7.3 Evaluate 3b + ab if a = -2 and b = 10. solution: First, replace a and b with the values shown. Any time you substitute a negative number, put it in parentheses. This will help you keep track of the negative sign. 3b + ab = 3(10) + (-2)10 = 30 + (-20) = 10 7B Basic Algebra Rules Working with : Any symbol can be used to represent a variable or unknown quantity, but in algebra courses, is the most-frequently used. If you haven t already, memorize the definitions, especially definitions regarding and eponents. Product and quotient rule for eponents: In Lesson 1, you learned some basics about eponents, such as 3 3 is the same thing as 3 2. We normally don t write eponents = 1, but if we do, we see that = =3 2. Replacing 3 with, we get 1 1 = 1+1 = 2. A good way to summarize the product rule is this: If you have similar bases multiplied together, simplify by adding the eponents. Considering division now, you know that = 3. Including eponents that equal 1, we have = = 3 1. Replacing 3 with, we get 2 1 = 2-1 = 1. A good way to summarize the quotient rule is this: If you have similar bases divided by each other, simplify by subtracting the eponents. Eample 7.4 Simplify a) 2 7 b) 2 7 c) y d) y 3 solution: a) 2 7 = 2+7 = 9 b) 2-7 = 2-7 = -5 c) Remember, only similar bases can be simplified: y = 3 9 y d) First, move the 3 to the numerator and make the eponent a y -3 Net, add eponents of similar bases together:

4 3 2+7+(-3) y = 3 9+(-3) y = 3 6 y Additive and multiplicative properties: These rules are the closest approimation to the original form of al-jabr discussed in 7A. The majority of problems you will solve in algebra are based on equations. Eample 7.5 Use the additive property to solve the following for. a) + 21 = 47 b) 38 = -7 + c) 1 + = solution: The additive property shows us that for any equation, we can add (or subtract) the same value from both sides of the equation, and it doesn t change the equation. The single biggest mistake students make on problems like this is they forget to add the same value to both sides. Don t forget! a) + 21= = 47 + ( 21) = 26 b) 38 = = 45 = = 45 c) = = = = 3 8 When solving algebraic equations like these, the goal is always to isolate the variable. In other words, don t stop until you have the variable by itself on one side of the equals sign or the other. It doesn t matter which side, but normally, the variable is put on the left side. Eample 7.6 Use the multiplicative property (and additive if necessary) to solve for. a) 3 = 12 b) 1 = 1 c) = solution: a) 3 12 = 3 3 = 4 b) invert and multiply: = = = For c), use the additive property first to get rid of the 5, then divide both sides by = = = 7 3 = 7 3 = 2 1 3

5 Practice Set 7 (subscripts tell you which lesson each problem came from) You may use a calculator for problem 13 only Evaluate 2b - a if a = -7 and b = Evaluate 3t + k 2 if t = -7 and k = Evaluate - y if = -2 and y = Simplify. 3 y Simplify. ab 3 a Simplify Use the additive and multiplicative properties to solve the following equations = = = = If Jerry skinned 162 chickens in 3 hours, what was his average hourly rate? The model car was a 1/16th scale replica. If the actual car was 18 feet long, how long was the model? Round answer to 2 decimal places Fill in the empty cells of the following table: Round missing decimal and percent values to 2 or 3 decimal places. You may use a calculator % of 48 is what number? Fraction Decimal Percent 1/ % Divide Which of the following is an irrational number? a) 2 b) 4 c) 0.3 d) Reduce Simplify. -(-(-3)) Simplify Find the product of 221 and 1.7