Variable and Expressions

Variables and Expressions Aim: – To translate between words and algebraic expressions. -- To evaluate algebraic expressions.

Vocabulary A variable is a letter used to represent an unknown quantity. Ex. x y z k m a A constant is a value that does not change. Ex. 5 – π A numerical expression may contain only constants, operations, and grouping symbols. Ex. 4 – ÷ 2

Vocabulary An algebraic expression (or variable expression) may contain anything in a numerical expression, but it also contains one or more variables. Ex. 3x² – 4x + 3 An equation states that two expressions are equal. Ex. 5x – 3 = 7x + 5 A term is a constant, variable, or product or quotient of a variable and a constant. Ex. 3 x 4y w/3

Vocabulary To evaluate an expression, condense it into a single value using the order of operations. – Algebraic expressions can only be evaluated if we know the value of all variables. To simplify an algebraic expression, combine all like terms and follow any applicable algebraic and exponent laws. To solve an equation, isolate the given variable on one side and simplify the other side.

Example Evaluate each expression. (3 + 9) ÷ 3 12 ÷ 3 Simplify the contents of the parentheses. 4 Divide ÷ Divide. 13 Add.

Example Evaluate each expression when a = 2 and b = –3. 4a + 5b 4(2) + 5(–3) Plug in the given values. 8 + (–15) Multiply. –7 Add. –2a – b – b² –2(2) – (–3) – (–3)² Plug in the given values. –2(2) – (–3) – 9 Evaluate the power. –4 – (–3) – 9 Multiply. –10 Subtract from left to right.

Translating into Algebra A large part of this course involves writing your own algebraic expressions or equations from word problems. The following information provides some hints for knowing which operation is implied in word problems. – Note: This information is not all-inclusive. Nor is it a guarantee of which operation to use (consider “per” and “each”). It simply lists common phrases for each operation.

Addition The sum is the answer to an addition problem. Key words / phrases: sum, increased by, put together, combine, more Example: x + 3 The sum of x and 3. Three more than x. x increased by 3.

Subtraction The difference is the answer to a subtraction problem. Key words / phrases: difference, less than, how much more, how much less, reduced by, fewer, decreased by Example: x – 7 The difference of x and 7. Seven less than x. x decreased by 7.

Multiplication The product is the answer to a multiplication problem. Key words / phrases: product, times, equal groups, put together, per, each Example: 4x The product of 4 and x. Four times x. Four equal groups of x.

Division The quotient is the answer to a division problem. Key words / phrases: quotient, divided by, split into equal groups, separated, per, each Example: x ÷ 6 The quotient of x and 6. x divided by 6. Six divided into x.

Example Jonathan reads 20 pages per hour. Write an algebraic model for the number of pages he reads in h hours. 20h Renequa is 3 years younger than José, who is j years old. Write an algebraic model for Renequa’s age. j – 3

Example You have $10. Write an algebraic model for the amount you have left if you buy b bottles of water that cost $0.85 each. $10 – $0.85b Paul can run one mile in 6 minutes. Write an expression for the number of miles that Paul can run in m minutes. m ÷ 6