Introduction to Variables, Algebraic Expressions, and Equations You Need Your Comp Book

What Is Algebra? b Algebra is a system that works from the known to the unknown. 2

A combination of operations on letters (variables) and numbers is called an algebraic expression. Algebraic Expressions 5 + x 6  y 3  y – 4 + x 4x means 4  x and xy means x  y 3

4 Algebraic Expressions Algebraic Expressions are not solved they are evaluated. Riemann hypothesis

Replacing a variable in an expression by a number and then finding the value of the expression is called evaluating the expression for the variable. 5

Evaluate x + y for x = 5 and y = 2. Evaluating Algebraic Expressions x + y = ( ) + ( ) Replace x with 5 and y with 2 in x + y. 52 = 7 6

Equation Statements like = 7 are called equations. An equation is of the form expression = expression An equation can be labeled as Equal sign left sideright side x + 5 = 9

Solving/Solution When an equation contains a variable, deciding which values of the variable make an equation a true statement is called solving an equation for the variable. A solution of an equation is a value for the variable that makes an equation a true statement.

Solving/Solution... Determine whether a number is a solution: Is -2 a solution of the equation 2y + 1 = -3? Replace y with -2 in the equation. 2y + 1 = -3 2(-2) + 1 = -3 ? = = -3 ? True Since -3 = -3 is a true statement, -2 is a solution of the equation.

Solving/Solution... Determine whether a number is a solution: Is 6 a solution of the equation 5x - 1 = 30? Replace x with 6 in the equation. 5x - 1 = 30 5(6) - 1 = 30 ? = = 30 ? False Since 29 = 30 is a false statement, 6 is not a solution of the equation.

To solve an equation, we will use properties of equality to write simpler equations, all equivalent to the original equation, until the final equation has the form x = number or number = x x = number or number = x Equivalent equations have the same solution. The word “number” above represents the solution of the original equation. Solving/Solution...

Keywords and phrases suggesting addition, subtraction, multiplication, division or equals. AdditionSubtractionMultiplicationDivision Equal Sign sumdifferenceproductquotientequals plusminustimesintogives added to less than ofper is/was/ will be more than lesstwicedivideyields total decreased by multiply divided by amounts to increased by subtracted from double is equal to 12

the product of 5 and a number 5x5x5x5x twice a number 2x2x2x2x a number decreased by 3 n - 3n - 3n - 3n - 3 a number increased by 2 z + 2z + 2z + 2z + 2 four times a number 4w4w4w4w Translating Word Phrases into Expressions 13

+ 7x x + 7 three times the sum of a number and 7 3( + 7) 3(x + 7) the quotient of 5 and a number the sum of a number and 7 Additional Word Phrases into Algebraic Expressions... 14

Helpful Hint Remember that order is important when subtracting. Study the order of numbers and variables below. PhraseTranslation a number decreased by 5 x – 5 a number subtracted from 5 5 – x 15

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Def: The whole numbers The whole numbers are the counting numbers: 0,1,2,3,4,5,6,7,8,9,10,11,12,13,…

Def: The number line The number line is a picture that represents the numbers: Numbers increase from left to right

Property: Inequalities On the number line, If a number lies to the right, it is greater If a number lies to the right, it is greater If a number lies to the left, it is less If a number lies to the left, it is less

4 lies to the right of 2

Notation: Symbols for Inequalities We use the symbol “>” to represent “greater than” We use the symbol “>” to represent “greater than” We use the symbol “<” to represent “less than” We use the symbol “<” to represent “less than” We read our mathematical sentences from left to right, just like in English. We read our mathematical sentences from left to right, just like in English.

Mathematical symbols representing “four is greater than two” are shown below: 4 > 2

Remember the Alligator Principle!!! 4 > 2

Def: Rounding Rounding is an important skill in our increasingly complex world. Rounding is an important skill in our increasingly complex world. The use of rounding allows us to better understand numbers and what they represent. The use of rounding allows us to better understand numbers and what they represent. Rounding also allows us to quickly understand the magnitude of complex- looking numbers. Rounding also allows us to quickly understand the magnitude of complex- looking numbers.

Examples of Rounding If we are talking about 6,345,989,857 people, it is easier to say (and understand) 6.3 billion people. If we owe $14,763.94, it is easier to say (and understand) $15,000.

Method: Rounding A Whole Number 1. Identify the round-off place digit (ones, tens, hundreds,…). 2. If the digit to the right of the round-off digit place is: a. Less than 5, do not change the round-off place digit. b. 5 or greater, increase the round-off place digit by Replace all digits to right of round-off place digit with zeros.

Examples

Def: Variable A letter that represents a number is called a variable. Ex: A number plus 7 equals 21. What is that number? Ans. We can represent the unknown number as “X” and write this question using mathematical symbols: X + 7 = 21, X = ??? We see that X = 14.

Def: Expression An expression is a collection of numbers, variables, and operations. 3x – 4, 9 ÷ , 6xy + 4z Expressions

Property: Identity Property of Zero Adding zero to anything doesn’t change the number. This property represented symbolically: 0 + X = X, X + 0 = X Ex:0 + 5 = 5, = 5

Property: Commutative Property of Addition Two numbers can be added in either order with the same result: a + b = b + a Ex: = Both equal 13!

Def: Simplifying When possible, it is good to make things more simple: can be rewritten as 13 We simplified by combining the two numbers, 9 and 4, into one single number, 13.

Example Simplify: X Ans: X = 10 + X Note we combined the two numbers into one. Answ er

Property: Associative Property of Addition When we add three or more numbers, the addition may be grouped in any way. (a + b) + c = a + (b + c) Ex: (4 + 9) + 1 = 4 + (9 + 1) = = 14

Example Simplify: (14 + X) + 15 Ans: (14 + X) + 15 = (X + 14) + 15 commutative = X + ( ) associative = X + ( ) associative = X + 29 = X + 29

Def: Evaluating an Expression To evaluate an algebraic expression, we replace the variables in the expression with their corresponding values and simplify.

Example Evaluate X + 2 given that X = 5. Ans: Replace X with 5 and simplify. X + 2 = = 7 Answ er

Example Evaluate a + b + 7 given that a equals 9 and b equals 13. Ans: Replace a with 9, b with 13, and simplify. a + b + 7 = = = 29

Property: Subtraction is Not Commutative 3 – 2 does not equal 2 – 3 “not equal to” sign

Section 1.4 Multiplying Whole Number Expressions

Property: Multiplying Multiplying can be thought of as repeated addition. Multiplying can be thought of as repeated addition. Four 8’s

Def: Area Area is derived using multiplication. Area is derived using multiplication. A square foot is defined as the area of a square whose sides are 1 foot long. A square foot is defined as the area of a square whose sides are 1 foot long.

Property: Area of a Rectangle If we think of a rectangle as being composed of these boxes, we see that the area is equal to If we think of a rectangle as being composed of these boxes, we see that the area is equal to Area = (Length) x (Width)

Def: Factors and Products Things that are multiplied together are called factors. Things that are multiplied together are called factors. The result of the multiplication is called the product. The result of the multiplication is called the product.

Property: Multiplication Multiplication is commutative: ab = ba Multiplication is commutative: ab = ba Multiplication is associative: (ab)c = a(bc) Multiplication is associative: (ab)c = a(bc) Identity property of 1: 1∙a = a∙1 = a Identity property of 1: 1∙a = a∙1 = a Multiplication property of 0: 0∙a = a∙0 = 0 Multiplication property of 0: 0∙a = a∙0 = 0

Numbers b Monomial b Any number, all by itself, is a monomial, like 5 or A monomial can also be a variable, like “m” or “b”. It can also be a combination of these, like 98b or 78xyz. b It cannot have a fractional or negative exponent. Ex 46

Numbers b Binomials b A binomial is an equation or expression with two terms. 3x + 1, 2x 3 - 5x, x 4 - 4, x - 19 are examples of binomials. As well as 3x-3 = 10 or 5x =9 5x =9 47

Numbers b Trinomials b A trinomial is a polynomial with three terms. Examples of trinomials are 2x 2 + 4x - 11, 4x x + 9, 7x x x, and 5x x