The Five Basic Skills of Algebra : Simplifying Evaluating Solving Factoring Graphing Do whatever you are allowed to do, according to the rules of algebra Substitute numbers for letters, then do the arithmetic Figure out what the letter must equal, by doing OPPOSITE operations, to both sides of the equation, in BACKWARDS order The reverse of the distributive property: divide the expression into factors (things that are multiplied) Express a relation or an equation as a picture, either on a number line or on a coordinate plane

Simplifying Combining like terms Multiplying and Dividing with Exponents The two Distributive Properties “Invisible 1s” The Subtraction Property

Simplifying: Combining “Like Terms” “Fundamental Rule of Addition”: You can only add together things of the same type. In arithmetic, this requires “common denominators” in fractions, and “lining up the decimal point” in decimals In algebra, like terms have the same letters and the same exponents

Simplifying: Combining “Like Terms”, continued: “6 apples plus 3 dollars” can’t be combined, because apples and dollars are not like terms, just like 6a + 3 3x + 5x = 8x -2p 3 – 5p 3 = -7p 3 (5m + 9r) + (3r + 11 – 4m) = m + 12r + 11

Simplifying: Exponent Rules Exponents show repeated multiplication, the number of times a factor is present. (4x 2 )(2x 3 ) means “two factors of x” in the first parenthesis, multiplied by “three factors of x” in the second parenthesis, or five (two plus three more) factors of x, multiplied together. 4x 2 2x 3 = 4xx2xxx = 42xxxxx = 8x 5 So, when the same letters that have exponents are multiplied, the exponents are ADDED, even though it is a multiplication problem

Simplifying: Exponent Rules, continued Exponents show repeated multiplication, the number of times a factor is present. (4x 3 ) 2 means “4x 3 ” times “4x 3 ”, which means 4xxx4xxx, which equals 44xxxxxx, which equals 16xxxxxx = 16x 6 16 = 4 2, and (x 3 ) 2 = x 6. The exponent is distributed over the factors (the OTHER Distributive Property) So, when letters that have exponents are raised to another exponent ON THE OUTSIDE OF THE PARENTHESIS, the exponents are MULTIPLIED, even though it is an exponent problem

Simplifying: The TWO Distributive properties Operations “distribute” over the operation(s) ONE LEVEL below: EXPONENTS distribute over MULTIPLICATION and/or DIVISION MULTIPLICATION & DIVISION distribute over ADDITION and/or SUBTRACTION BUT…. EXPONENTS DO NOT distribute over ADDITION & SUBTRACTION ! ORDER OF OPERATIONS : (PARENTHESES override the order) Exponents and/or Radicals Multiply and/or Divide, L→R Add and/or Subtract, L→R

Simplifying: The TWO Distributive properties, continued EXAMPLES: A) 3(2e + 5m – 7) = 3 times 2e, plus 3 times 5m, minus 3 times 7, which equals 6e + 15m – 21 B)(5p 2 r) 3 = 5 to the 3 rd power, times p 2 to the 3 rd power, times r to the 3 rd power, which equals 125p 6 r 3 C)(3x + 5) 2 does NOT equal (3x) , and it does not equal 9x , either. EXPONENTS DO NOT DISTRIBUTE OVER ADDITION OR SUBTRACTION !

Simplifying: Invisible 1’s In Arithmetic, there are some instances where the number “1” can be present, but is often “invisible”. Fractions : 6 = 6 / 1 (fractions really mean division) Multiplication : 15 = 15 x 1 The same concept applies in Algebra. There are some instances, also dealing with multiplication or division concepts, where the number “1” can be present, but is often “invisible”. Coefficients : m = 1m = 1 m Parentheses : 9 – ( 2x + 5) = 9 – 1( 2x + 5) = 9 – 1 ( 2x + 5) Exponents : 5r = 5r 1, or 5 1 r 1, or (5r) 1

Simplifying: Subtraction Property Subtraction can be defined as “Addition of the Opposite Number”. So, anytime you have a subtraction problem, you can make TWO changes to turn it into an ADDITION problem : 1)Change the subtraction sign into an addition sign 2)Change the second number into its (pos. or neg.) opposite So, 5x – 3y can become 5x + ‾ 3y, and ( 5 + 2x ) – ( 7 – 6x ) can become (5 + 2x ) + ‾(7 + ‾ 6x), which (because of invisible 1’s and distributing) can become ( 5 + 2x ) + ‾1 ( 7 + ‾ 6x ), = 5 + 2x + ‾ 7 + 6x, or ‾ 2 + 8x

For practice, Simplify each expression: 1)8x – 3y + 2 ( 9x – 5y + 4 ) 2)(3x 4 y 5 ) x 4 y 5 = 3)( 5x – 7 ) 2 = 4)2( 3x + 4y – 5 ) – ( ‾ 7x – 10y + 12 )

Evaluating In evaluating, you will be told what number value each of the letters equal. You will need to figure out the value of the entire expression. There are two steps: 1)Re-write the expression, substituting the numbers for the letters 2)Do the arithmetic, following the order-of-operations rules [ PERMDAS, Parenthesis, Exponents, Radicals, Multiplication, Division, Addition, Subtraction ] Example: Evaluate 3x 2 + 2xy – 8y + 9, if x = ‾ 3, y = 5 1) 3 (-3 ) – ) – ) 27 + ‾ 30 – ) 36 + ‾ 70 = ‾ 34

For practice, evaluate each expression: 1)2x – 7 if x = – 3 2)8xy – 3y + 2 ( 9x – 5y + 4 ) if x = 3, y = – 4 3)x 2 + 4x + 3 if x = 4 4)( 5x – 7 ) 2 if x = 2.5

Solving Solving is the opposite of evaluating. In solving, you will be told the value of the entire expression. You will need to figure out what number value each of the letters equal. There are two basic concepts: 1)“Undo” any operations by doing the ‘opposite” operation to both sides of the equation 2)Perform the opposites in the REVERSE of the order-of- operations rules [ PERMDAS ]. So, get rid of Addition or Subtraction first, then get rid of Multiplication or Division, then get rid of Exponents or Radicals. For practice, solve these: 1)2x – 8 = -142) 3x – 12 = 63) 2

Factoring Factoring is the process of changing an expression with TERMS (things that are added or subtracted) into an expression of FACTORS (things that are multiplied). Instead of ___ + ____ – ___, you get ( ___ )( ___ )( ___ ) There are three basic types of factoring: 1)Factoring out a GCF, for example, 6x 4 y 2 + 9x 3 y 5 can be factored into 3x 3 y 2 ( 2x + 3y 3 ) 2)Factoring a Quadratic Trinomial for example, 3x 2 – 7x + 4 can be factored into (3x – 4) (x + 1) 3)Special Factoring Patterns, which should be memorized For example, 25x 2 – 9y 2, or 4x 2 – 20x + 25, or y 3 + 8b 3 Factoring can be thought of as the “reverse” of the (first) Distributive property. If you distribute your factored answer, you should get the original expression.

For practice, factor each expression: 1)24x 4 – 36x 2 – 48x 2)6x 2 + 9x + 10x – 15 3)x 2 + 4x + 3 4)3x 2 – 11x + 6

Graphing Graphing is the process of creating a mathematical picture of a relationship between variable amounts. Just as it is often easier to get information from a picture than from a written description, it is also often easier to get information from a graph than from an algebraic relationn. There are two major types of graphs: Graphs on a Number Line, usually used for inequalities in ONE variable, Graphs on a Rectangular Coordinate Plane (AKA an “x & y axis” or a “Cartesian Plane”, used for equations and inequalities in TWO variables (usually “x” & “y”)

For practice, graph the following, using any method 1) y = x 2 – 5x + 6 2) y > ½x – 3 3) y = 7