Exam Review Christian Martin Algebra (6th period) Ms.Hardtke 5/17/10
Addition Property of Equality if the same number is added to both sides of an equation, the two sides remain equal. Ex. If X=Y, then X+Z=Y+Z
Multiplication Property of Equality • Is when both sides of an equal equation is multiplied and the equation remains equal • Ex.
Reflexive Property of Equality The reflexive property of equality says that anything is equal to itself For example A=A or, 145=145
Symmetric Property of Equality Symmetric property of equality is two variables that are different but have the same number/amount (equal symmetry) Ex. If a=b, then b=a
Transitive Property of Equality When numbers or variables are all equal Ex. If a=b and b=c, then c=a If A=3.3 and 3.3=B, then A=B
Associative Property of Addition The sum of a set of numbers is the same no matter how the numbers are grouped. For example: 2+(3+Y)= (2+3)+Y= 5+Y
Associative Property of Multiplication The product of a set of numbers is the same no matter how the numbers are grouped. For example 5(4a)=(5x4)a=20a by using the associative property we switch the 5 and 4
Commutative Property of Addition The sum of a group of numbers is the same regardless of the order in which the numbers are arranged, for example: 3+7=7+3 X+2 = 2+X
Commutative Property of Multiplication The product of a group of numbers is the same regardless of the order in which the numbers are arranged, for example: 3xA=Ax3 Xy=yx
Distributive Property Is distributing something as you separate or break it into parts. Ex. A(b+c)= Ab + Ac 2x(3y+8)=6xy + 16x
Property of Opposites/Inverse Property of Addition A number added to its opposite integer will always equal zero. Ex. A+(-A)=0 3+(-3)=0 or (-3)+3=0
Property Of Reciprocals/Inverse Property of Multiplication For two ratios, if a/b =c/d, then b/a = d/c a(1/a) = 1 A number times its reciprocal, always equals one A Reciprocal is its reverse and opposite (the signs switch from + to — or vice versa)
Identity Property of Addition Identity property of addition states that the sum of zero and any number or variable is the number or variable. 4+0=4, -11+0=-1, y+0=y
Identity Property of Multiplication Identity property of multiplication states that the product of 1 and any number or variable is the number or variable itself 4×1=4, -11×1=-11, y×1= y
Multiplicative Property of Zero The product of any number and zero is zero Ax0=0
Closure Property of Addition Closure property of addition states that the sum of any two real numbers equals another real number. 2 + 5 = 7
Closure Property of Multiplication Closure property of multiplication states that the product of any two real numbers equals another real number. Ex. 4 × 7 = 28
Product of Powers Property Is that when you multiply powers having the same base, add the exponents. Ex. am × an = am+n.
Power of a Product Property This property states that the power of a product can be obtained by finding the powers of each factor and multiplying them. Ex. (ab)m = am × bm
Power of a Power Property the power of a power can be found by multiplying the exponents. Ex. (am)n = amn
Quotient of Powers Property This property states that to divide powers having the same base, subtract the exponents.
Power of a Quotient Property This property states that the power of a quotient can be obtained by finding the powers of numerator and denominator and dividing them.
Zero Power Property If a variable has an exponent of zero, then it must equal one 30=1
Negative Power Property When a fraction or a number has negative exponents, you must change it to its reciprocal in order to turn the negative exponent into a positive exponent 2-2 =1/2 2 = 1/4
Zero Product Property When both variables equal zero, then one or the other must equal zero Ex. if ab=0, then either a=0 or b=0 if XY = 0, then X = 0 or Y = 0 or both X and Y are 0.
Product of Roots Property The product is the same as the product of square roots
Quotient of Roots Property the quotient is the same as the quotient of the square roots
Mini Quiz • What Property of Equality states that anything is equal to itself? A) Reflexive B) Symmetric C) Multiplication D) None A) Reflexive [remove for answer]
Solving 1st power inequalities in one variable Here is a table explaining inequalities Also when dividing by a negative, the inequality sign must be switched in order for the statement to be true
Inequalities cont. • A conjunction is true only if both the statements in it are true • A conjunction is a mathematical operator that returns an output of true if and only if all of its operands are true. • Ex. -2<x and x<3 -3 -2 -4 0 1 2 3
Ineq. Cont. Adisjunction is statement which connects two other statements using the word or. To solve a disjunctions of two open sentences, you find the variables for which at least one of the sentences is true. The graph consists of all points that are in the graph Ex. -2<x or x<3 -3 -2 -1 0 1 2 3
Mini Quiz • Solve the inequality: x-1>4 Answer: x>5
Linear Equations Solving equations in two variables Graphing points Standard/General Form Slope- Intercept Form Point-Slope Form Slopes
Solving equations in two variables the solutions for an equation in two variables are ordered pairs in the form (a, b). Ex. x = y + 1 is true when x = 3 and y = 2. So, the ordered pair (3, 2) is a solution to the equation.
Graph Points A Graph Point contain an X and a Y The X-number line crosses horizontally and the Y-line vertically. Y line graph X line graph
Standard/ General Form Standard/ General form Ax + By = C The terms A, B, and C are integers (could be either positive or negative numbers or fractions)
Slope-Intercept Form where m is the slope of the line and b is the y-intercept, which is the y-coordinate of the point where the line crosses the y axis. This can be seen by letting x = 0, which immediately gives y = b. Vertical lines, having undefined slope, cannot be represented by this form. The graph of this equation is a straight line. The slope of the line is m. The line crosses the y-axis at b. The point where the line crosses the y-axis is called the y-intercept. The x, y coordinates for the y-intercept are (0, b).
Point-Slope Form where m is the slope of the line and (x1,y1) is any point on the line. The point-slope and slope-intercept forms are easily interchangeable. The point-slope form expresses the fact that the difference in the y coordinate between two points on a line (that is, y − y1) is relative to the difference in the x coordinate (that is, x − x1). The proportionality constant is m (the slope of the line).
Slopes • To find a slope use this formula: • m= X2 - X1 Y2-Y1
Slope of a Straight Line One of the most important properties of a straight line is in how it angles away from the horizontal. This concept is reflected in something called the "slope" of the line. To find the slope, you will need two points from the line and you will get these points by doing the point-slope formula. Then use the formula from the previous slide.
Mini Quiz • What is the equation for the Point-Slope Form? Answer:
Linear Systems Substitution Method Addition/ Substitution Method (Elimination)
Substitution Method Ex. X+Y=2 Y= 2x-1 substitutes for the Y in the equation above making it X+(2x-1)=2 3x-1=2 3x=3 X=1 then plug the x-coordinate into the equation to find the y-coordinate Y=2(1)-1 Y=2-1 = 1 thus the answer will be (1,1)
Addition/Subtraction Method 1. Algebraically manipulate both equations so that all the variables are on one side of the equal sign and in the same order. (Line the equations up, one on top of the other.) 2. If needed, multiply one of the equations by a constant so that there is one variable in each equation that has the same coefficient. 3. Subtract one equation from the other. 4. Solve the resulting equation for the one variable. 5. Using the value found in the step 4, substitute it into either equation and solve for the remaining variable. 6. Substitute the values for both variables into the equation not used in step 5 to check.
Factoring Greatest Common Factor (GCF) Difference of Squares Sum and Difference of Cubes PST Reverse Foil Grouping 2x2
GCF • Ex. 3x3 + 27x2 + 9x • To factor out the GCF in an expression like the one above, first find the GCF of all of the expression's terms. • 3(1, 3) • 27(1, 3, 9, 27) • 9(1, 3, 9) • Next, write the GCF on the left of a set of parentheses: 3x( ) • Next, divide each term from the original expression (3x3+27x2+9x ) by the GCF (3x), then write it in the parenthesis =3x(x2 + 9x + 3)
Difference of Squares When the sum of two numbers multiplies their difference then the product is the difference of their squares. Symmetrically, the difference of two squares can be factored: x² − 25 = (x + 5)(x − 5) x² is the square of x. 25 is the square of 5.
Sum & Difference of Cubes A polynomial in the form a3 + b3 is called a sum of cubes. A polynomial in the form a3 − b3 is called a difference of cubes.a3 + b3 = (a + b)(a2 – ab + b2) a3 – b3 = (a – b)(a2 + ab + b2)
PST A perfect square trinomial is: (ax)2 + 2abx + b2 Recognizing it can save a lot of time Ex. x2 + 2x + 1 = 0 (x + 1)2 = 0 X+1=0 x= -1