Algebra Basics

Variables and Expressions Variable – a symbol used to represent a quantity that can change. Coefficient – the number that is multiplied by the variable in an algebraic expression. Numerical expression – an expression that contains only numbers and operations. Algebraic expression – an expression that contains numbers, operations and at least one variable. Constant – a value that does not change. Evaluate – To find the value of a numerical or algebraic expression. Simplify – perform all possible operations including combining like terms. Order of operations – a rule for evaluating expressions from left to right.

Order of Operations When a numerical expression involves 2 or more operations, there is a specific order in which these operations must be performed. In earlier grades you learned to solve numerical expressions using “order of operations” or as most of us say PEMDAS which stands for Parenthesis, Exponents, (Multiplication/Division) then (Add/Subtract) PEMDAS Please Excuse (My Dear) (Aunt Sally)

Please Excuse (My Dear)(Aunt Sally) The reason (multiplication & division) and (add & subtract) are grouped is when those operations are next to each other you do the math from left to right. You do not necessarily do addition first if it is written next to subtraction.

equation – is a mathematical statement that two expressions are equal (=). formula – is an equation that shows a mathematical relationship between 2 or more quantities. reflexive property of equality – for all real numbers a, a = a. symmetric property of equality – for all real numbers a and b, if a = b, then b = a. transitive property of equality – for all real numbers a, b, and c, if a = b and b = c, then a = c

Numerical expression 8 – 6 + 2 Subtraction is first Remember when those operations, (multiplication & division) and (add & subtract) are grouped next to each other you do the math from left to right.

2 + 3[5 +(4-1)²] 2 +3[5 + (3)²] 2 + 3[5 + 9] 2 + 3[14] 2 + 42 44 When there are 2 or more operations, and we use grouping symbols such as parenthesis or brackets, you perform the inner most grouping symbol first. 2 + 3[5 +(4-1)²] 2 +3[5 + (3)²] 2 + 3[5 + 9] 2 + 3[14] 2 + 42 44

Summary of Basic Steps Copy the problem. Simplify any grouping symbols (such as parenthesis) first, starting with the inner most group. Simplify any powers (exponents). Perform the multiplication & division in order from left to right. Do the addition & subtraction last, from left to right. Remember, if operations are written next to each other, work from left to right.

It is very important to understand that it does make a difference if the order is not performed correctly!!!!!!!!!

Order of Operations is not an isolated skill. This skill applies to almost every topic in Math. Remember that “Aunt Sally” is used with evaluating formulas, solving equations, evaluating algebraic expressions, simplifying monomials & polynomials, etc…

Order of Operations Get your pencil and calculator ready and try these problems. 20 + 3(5 – 1) = 3 + 2²(1 + 8) = (5 · 4)² = 4) 2(3 + 5) – 9 = 5) 2[13 – (1 + 6)] = 48 / 3 + 5 = 7) 3(6 + 4)(5 – 3) = 8) 100 – 4(7 – 4)³ = 9) 1² + 2³ + 3³ = 10) (24 – 6)/2 =

Here is an example of an algebraic expression. 4 is the coefficient Here is an example of an algebraic expression. 4 is the coefficient. X is the variable. 7 is the constant.

Evaluating Algebraic Expressions To evaluate an algebraic expression, substitute a given number for the variable, and find the value of the resulting numerical expression. X – 5 for x = 12 (12) – 5 = 7 2y + 1 for y = 4 2(4) + 1 = 9 6(n + 2) – 4 for n = 5, 6, 7 38, 44, 50

Evaluate each expression for t = 0, x = 1.5, y = 6, z = 23 2) 3z – 3y 3) z – 2x 4) xy 5) 4(y – x) 6) 3(4 + y) 7) 3(6 + t) – 1 8) 2(y – 6) + 3 9) y(4 + t) - 5 10) x + y + z

Properties of Real Numbers