Section Introduction to Algebra: Variables & Mathematical Models

Transcription

1 Math Section Introduction to Algebra - Page 1 Section Introduction to Algebra: Variables & Mathematical Models I. Basic Definitions A. A variable is a letter. We can assign any number to the variable. So the value of the variable can change or vary. B. A constant is a number. Its value does not change but stays constant. C. An algebraic expression is a combination of variables and constants with basic mathematical operations such as addition, subtraction, multiplication or division. D. In algebra, we denote multiplication in several ways: 1. (2)(x) means two times x x also means two times x. 3. 2(x) means two times x. 4. 2x also means two times x. E. Whenever we write a constant in front of a variable with no operation written, it means multiply. F. The constant in front of the variable is called the coefficient. II. Evaluating Algebraic Expressions A. When we evaluate an algebraic expression (usually we just say evaluate the expression), we have to substitute, or replace, the variable with a constant and perform the indicated mathematical operations. B. Remember that you have to use the order of operations. 1. Parenthesis 2. Exponents 3. Multiplication/Division left to right 4. Addition/Subtraction left to right 5. PEMDAS! C. Examples - Evaluate each expression for x = x + 10 We begin by substituting 4 in for x Now add Answer: x Again, we substitute 4 in for x. Remember that "5x" means "5 times x" (4) Multiply 5 and Add. Answer: 23 2x8 2x3. Now you try one: Answer: 5

2 (())Math Section Introduction to Algebra - Page 2 D. Examples - Evaluate each expression for x = 7 and y = (x + y) We substitute 7 in for x and 5 in for y. 3(7 + 5) Add 7 & 5 in the parenthesis. 3(12) Multiply. Answer: 36 2y 2xy2. xeverywhere there is an x, substitute in a Everywhere there is a y, substitute in a Simplify the top and the bottom. 97Divide. The fraction bar is considered a grouping symbol. So we need to simplify the top and the bottom first. On the top, multiply the 2 & 5 first. On the bottom, multiply the 2 & 7 first. 5Answer: 3 21x y3. Now you try one: Answer: 10 35III. Translating from English into Algebra A. There are several buzz words that, when you see them in an English phrase, they have a definite mathematical meaning. B. Addition 1. add 2. sum 3 increased by 4. more than C. Subtraction 1. subtracted from ***** 2. subtracted by 3. decreased by 4. less than ***** 5. difference 6. ***** = Don't forget to switch the order! D. Multiplication 1. times 2. multiply by 3. product 4. of 5. twice = 2 x 6. triple = 3 x

3 Math Section Introduction to Algebra - Page 3 E. Division 1. divided by 2. quotient 3. any fraction! F. Equality 1. is 2. was 3. were 4. Any form of the verb "to be". 5. gives G. Examples - Write each English phrase as an algebraic expression. Let x represent the number. 1. six more than a number "more than" means addition. Answer: 6 + x 2. six less than a number 'less than" means subtract, change the order! Answer: x 6 3. Now you try one: the sum of a number and 4. Answer: x three subtracted from a number "subtracted from" means subtract, change the order! Answer: x 3 5. three decreased by a number "decreased by" means subtract Answer: 3 x 6. Now you try one: three times a number, decreased by 5. Answer: 3x 5 7. The sum of 20 divided by a number and that number divided by 20. "sum" means add. But what are we adding? We are adding 20 divided 20x x 20 by a number and the number divided by 20 20x. x 20Answer: 8. Five times a number is 35. "times" means multiply; "is" means equals. Answer: 5x = 35

4 Math Section Introduction to Algebra - Page Now you try one: The quotient of 14 and a number is 141 x2answer: The sum of twice a number and 6 is 16. "sum" means add; "twice" means two times; "is" means equals. Answer: 2x + 6 = Three less than 4 times a number gives 29. "less than" means subtract, change the order!; "times" means multiply; "gives" means equals. Answer: 4x 3 = Now you try one: Five times a number is equal to 24 decreased by the number. Answer: 5x = 24 x IV. What is the solution? A. A solution to an equation is any number, that when substituted for the variable, after evaluation, gives us a true statement. B. Examples - Determine whether the given number is a solution of the equation. 1. x + 17 = 22;5 Substitute 5 in for x = 22 Add the left hand side. 22 = 22 True. Answer: It is a solution. 2. 5z = 30;8 Substitute 8 in for z, remember that 5z means "5 times z". 5(8) = 30 Multiply the left hand side. 40 = 30 False. Answer: It is not a solution. 46 r3. Now you try one: Answer: It is a solution. 8;4. 3m + 4 = 19;6 Substitute 6 in for m; remember that "3m" means "3 times m". 3(6) + 4 = 19 Multiply 3 & = 19 Add 18 & = 19 False. Answer: It is not a solution (w + 2) = 4(w 3);10 Substitute 10 in for w on both sides. 3(10 + 2) = 4(10 3) Simplify in each parenthesis.

5 Math Section Introduction to Algebra - Page 5 3(12) = 4(7) Multiply each side. 36 = 28 False. Answer: It is not a solution. 6. Now you try one: 6(p 4) = 3p;8 Answer: It is a solution. V. Applications A. In solving word problems, one of the most important considerations is understanding what each variable represents. B. Use the information given to decide what variable to replace with the number given. C. Example - The bar graph (left-hand column, page 12) shows the average price of a movie ticket for selected years from 1980 through Here is a mathematical model that approximates the data displayed by the bar graph: T = 0.15N Average movie ticket price Number of years after Use the formula to find the average ticket price 5 years after 1980, or in Does the mathematical model underestimate or overestimate the average ticket price shown by the bar graph? By how much? (page 12, #84a) If 1980 is our base year, then 1985 means that N = 5 ( = 5). So substitute 5 in for N in the formula. T = 0.15(5) T = Multiply. Add. T = 3.47 Answer: The average ticket price was $3.47. Looking at the bar graph, the model underestimates the actual value by $0.08.

6 Math Section Introduction to Algebra - Page 6 2. Now you try #84b. Does the mathematical model underestimate or overestimate the average ticket price show by the bar graph for 2005? By how much? Answer: The average ticket price in 2005 was $$6.47. The mathematical model overestimates the bar graph by $0.06.