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Thinking Mathematically By: Mac Sinsky and Matt Hipp Solving 1st power equations in one variable A. Special cases where variables cancel to get all reals Example: 2x+6=2(x+3) -Distribute the 2 2x+6=2x+6 Cancel out to equal no set B. Equations containing fractional coefficients Example: x+ 10 =5 5 10 5 =2, so x=3 Equations with variables in the denominator Example: 28 7 x 28 divided by 7 =4, so x=4 In the next slides you will review: All the Properties, and then take a Quiz on identifying the Property Names Addition Property (of Equality) Definition: If one number is added to two sides of an equation Example: If 3=3. then 2 + 3=3 + 2 Multiplication Property (of Equality) Definition: If one number is multiplied to two sides of an equation Example: If 2=2, then 2 x 5=5 x 2 Reflexive Property (of Equality) Definition: One number equals the same number Example: 10=10 Symmetric Property (of Equality) Definiton: One number equals the same number, even if it’s in a different order Example: If 6=8, then 8=6 Transitive Property (of Equality) Definition: One number equals another number, and that number equals another number Example: If 12=15, and 16=12. then 15=16 Associative Property of Addition Definition: When three numbers are added, and changing the order has the same end result Example: (5 + 6)+ 4=5+(4 + 6) Associative Property of Multiplication Definition: When three numbers are multiplied, and changing the order has the same end result Example: (4 x 7)x 2=4 x(2 x 7) Commutative Property of Addition Definition: When two numbers are added, and, if the order is changed, the result is the same Example: 7 + 6=6 + 7 Commutative Property of Multiplication Definition: When two numbers are multiplied, and, if the order is changed, the result is the same Example: 9 x 8=8 x 9 Distributive Property (of Multiplication over Addition) Definition: When the distributive property equals the distributive property broken down Example: 4x(5+8)=4x5 +4 x 8 Prop of Opposites or Inverse Property of Addition Definition: When a number added to its opposite equals zero Example: 7 +(-7)=0 Prop of Reciprocals or Inverse Prop. of Multiplication Definition: when two numbers are multiplied, and the answer is 1 Example: 4 x ¼=1 Identity Property of Addition Definition: When 0 plus a number equals the number that was added to 0 Example: 0 + 5=5 Identity Property of Multiplication Definition: When 1 times a number equals the number that was multiplied by the 1 Example: 1 x 4=4 Multiplicative Property of Zero Definition: When 0 times a number equals 0 Example: 0 x 3=0 Closure Property of Addition Definition: When two numbers are added to equal a different number Example: 5 + 3=8 Closure Property of Multiplication Definition: When two numbers are multiplied to equal a different number Example: 4 x 6=24 Product of Powers Property Definition: When two powers and their exponents are added Example: 22 x 25=4 x 32=128 is the same as 22 + 5 =27=128 Power of a Product Property Definition: When you want to find the power of a product Example: (3 × 4)2 = 122 = 144 is the same as 32 × 42 = 9 × 16 = 144 Power of a Power Property Definition: When you want to find the power of a power Example: (22)3 = 43 = 64 is the same as 22×3 = 26 = 64 Quotient of Powers Property Definition: When dividing two powers that have the same base, subtract the exponents Example: 6 4 = 642 62 Power of a Quotient Property Definition: When dividing two bases with the same exponents, cancel out the common factors Example: ( 2 )2 22 4 is the same as 22 4 4 2 22 4 Zero Power Property Definition:When zero is the exponent the answer is 1 Example: 80 1 Negative Power Property Definition: When the power is a negative, the answer is 1 over that power. Moving the power to the denominator takes away the negative. Example: 94 1 4 Zero Product Property Definition: When two numbers equal zero, one of the numbers must be equal to zero Example: If ab=0, then a=0 or b=0 Product of Roots Property Definition: When two numbers, both inside the root, equal both numbers in separate roots Example: (5)(6) = 5x 6 Quotient of Roots Property Definition: When a fraction is in a root, simplify the numbers. 8 1 Example: 16 2 Root of a Power Definition: When an exponent is in a square root Example: 3 2 -The power and the root cancel, so final answer is 2 Power of a root Definition: When the square root is powered 2 Example: 5 - Multiply with the power and the number, then use the root 25 5 Now you will take a quiz! Look at the sample problem and give the name of the property illustrated. Click when you’re ready to see the answer. 1. 30 + 2 = 2 + 30 Answer: Commutative Property (of Addition) Now you will take a quiz! Look at the sample problem and give the name of the property illustrated. Click when you’re ready to see the answer. 2. If 65=54, and 25=65, then 25=54 Answer: Transitive Property (of Equality) Solving 1st power inequalities in one variable A. With only one inequality sign Example:2+x > 15 16>15, so 2+14>15 B. Conjunction Example:3 < x+15 < 20 18 is greater than 3, but less than 20, so 3 < 3+15 < 20 C. Disjunction Example:x+2 > 40 45 > 40, so 43+2 > 40 Linear Equations in Two Variables y 2 y1 x 2 x1 slopes of all types of lines: equations of all types of lines Example: y=3x+5 standard/general form: Ax+By=C point-slope form: (y-y1)=m(x-x1) how to graph: Use slope-intercept form (y=mx+b) Example: y=3x+5 (5,0) is y-intercept, and 3 is slope up (3,1) from y-intercept Linear Equations in Two Variables (cont.) how to find intercepts: find intercepts by using slope-intercept form: y=mx+b how and when to use the point-slope formula: The point-slope formula= y y1 m( x x1 ) Linear Systems Substitution Method Definition: When you subtract one side of an equation with another to get an answer Example: 5x+y=10, x+4y=10 --Get y alone, solve for x 1. y=10-5x, x+4(10-5x)=10 2. x+40-20x=10, -19x=10-40 3. y=10-5(30) y=-140 Linear Systems (cont.) Addition/Subtraction Method (Elimination) Definition: When you add two equations to find the variable. Example: 5x+4y=4, 4x-4y=5 -Add these two equations 1. 9x=9 or x=1 -Plug in x to first equation to find y 2. 5(1)+4y=4, 4y=-1 Y=-1/4 Linear Systems (cont.) Check for understanding of the terms dependent, inconsistent and consistent Dependent: When two lines are on top of each other Example: Consistent: When two lines cross once Example: Linear Systems (cont.) Inconsistent: When two lines are parallel Example: Factoring Sum and Difference of Cubes: 27 x3 125 y 3 Break the factors down: (3x 5x)(9 x2 15xy 25 y 2 ) Difference of Squares: 81a 2 25b 2 Break the factors down: (9a)2-(5b)2 (9a-5b)(9a+5b) Grouping 3x1: (x2+10x+25)-y2 Break the factors down: (x+5)+y(x+5)-y GCF: 16x5-20x4+8x3 Find GCF: 4x 3 Solve: 4x3(4x2-5x+2) Factoring (cont.) Reverse FOIL Example: (x+5)(5+x) 2 Solve: 5 x 25 x 5 x PST -PST stands for perfect square trinomial 2 Example: (2 x 9s) -Multiply using the power - Multiply the number 9 times the number of the power(2) Solve: 4r 2 +36rs+81s Grouping 2x2 Example: (a-9) 2 -Multiply with the power(2) -Multiply 9 with the number of the power(2) Solve: a 2 -18a+81 Rational Expressions A. Simplify by factor and cancel 3a 6 Example: 3a 3b -factor, then cancel out common factors 33aa36b 3(3(aa b2)) a b Rational Expressions (cont.) Addition and subtraction of rational expressions Example: 3c 5c 16 16 -Add numerator, and cancel 3c 5c 8c c 16 16 2 Rational Expressions (cont.) Multiplication of rational expressions Example: 4 x 21 7 8 -Multiply, and simplify 4 21 84 3 x 7 8 42 2 Quadratic equations in one variable Factoring Example for any terms: 5 x 2 15 x x=3 Example for binomials: x 2 25 X=5 Example for trinomials: x 2 8 x 16 -Put zero on one side, then find PST x 2 8 x 16 0 PST=-4 Quadratic equations in one variable (cont.) Square root of both sides Example: x 2 64 X=8 Complete the square Example: 3x 2 6 x 2 12 x - Get zero on its own side, then simplify 2 x 2 12 x 6 0 x 2 6 x 9 12 ( x 3)( x 3) x 3 12 x 3 2 3 Functions A. What does f(x) mean?: f(x)=y B. Find the domain and range of a function Domain: to see how much of x is covered Range: to see how much of y is covered Example: (1,1)(4,4)(5,3) Domain is 4- From 1 to 5 Range is 3- From 1 to 4 Functions (cont.) Given two ordered pairs of data, find a linear function that contains those points Example: (3,5)(7,10) -Find the slope 10 5 5 73 4 -Put slope in point-slope formula y 5 5 ( x 3) 4 5 15 x 4 4 5 35 y x 4 4 y 5 Functions (cont.) Quadratic functions Formula: y ax 2 bx c Example: y x 2 2 x 8 Because a=1, graph opens up To find x-intercept, set y to 0 0=(x-4)(x+2)=(4,0)(-2,0) To find y-intercept, set y to 0 f (0) (0)2 2(0) 8 (0, 8) Vertex: formula= 2 (1, 9) 2(1) b 2a Axis of symmetry: x=1 The discriminant ( b2 4ac ) tells you the number of x-intercepts. Simplifying Expressions with Exponents When two exponents are multiplied, add 2 2 4 ( x )( x ) x Example: When an exponent is multiplied by a power, multiply the powers Example: ( x 2 )3 x 6 If an exponent has a 0 power, the answer is 1 Example: x 0 1 When an exponent has a negative power, the answer is a fraction 1 2 Example: x 2 x Simplifying Expressions with Radicals When square root has a second power, the square root and power cancel out 2 Example: x 2 (15)2 x+2=225 x=223 Word Problems Mike spent $95 on Packer tickets. This was $25 less than twice of what John spent. How much were John’s tickets? 2x-25=95 2x=70 X=35 John’s Packer tickets cost $35. Word Problems (cont.) There are n number of players on a basketball team. This is four less than two times the amount of coaches. How many coaches are on the team? 2x-4=n 2x=b+4 X=2 There are two coaches. Word Problems (cont.) The sum of two consecutive numbers is 21. What are the numbers? 2x=21-1 2x=20 X=10 The numbers are 10 and 11 Word Problems (cont.) Two students are running for class president. One student got 30 more votes than the other student. If the total amount of votes is 45, how much votes did each person get? x+x+30=35 One x is bigger than the other x, so the first x is 10, and the second x is 5 Line of Best Fit or Regression Line When do you use this?: Linear regression on the TI-84 calculator is used when you want to find the line of best fit, or when you want to see the stat plots. You can also see both of these together. How does your calculator help?: Your TI-84 calculator can find out and graph your line of best fit Example: (-18,4)(-11,7)(-2,12.5)(1,14)(6,16)