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Beware of the Sucker Answer Make sure you answer the question that is asked! Double check the question before you fill in the bubble!! Do the Easy Ones First Then go Back and do the Hard Ones! For Multiple Choice Tests •Check each answer – if impossible or silly cross it out. •Back plug (substitute) – one of them has to be the answer •For factoring – Work the problem backwards •Sketch a picture •Graph the points •Use the y= function on calculator to match graphs Read the directions for the test carefully. Read each question carefully. Number next to a Letter (variable) means Multiply 4x 12a + -2b If x = -3 If a = 5, b = -3 Then substitute & multiply Then substitute & multiply 4(-3) = -12 12(5) + -2(-3) = 66 So do numbers next to ( ) and letters next to letters x·y = xy 4(x) = 4x a·b = ab (-2)(a) = -2a 4·f ·g = 4fg Addition and Subtraction are snobs. They just combine with their own kind. They form cliques. 3x – 3a + 7 + 6x + 2a = 9x – a + 7 4x² + 6x +9 – 15x + 3x² + 10 = 7x² - 9x + 19 That’s Just How They Do. That’s How It Is. Deal With It! Multiplication and Division are party animals. They will do it with anyone! 2x · 4y = 8xy a· b·c = abc 12xy/4x = 3y Understanding Algebraic Culture is the Key to Success!! 2=2÷3 3 Numbers come with Signs (+ -) The sign is in front Remember: These are the same thing 5–3=2 5+-3=2 Because Subtraction is Adding the Opposite If you are confused, Circle the number & the sign in front, then do the math! 2 – 56 + 7 – 8 – 10 = - 65 ABSOLUTE VALUE The absolute value is always positive. The absolute value of │5│ is 5 The absolute value of │-5│ is 5 To solve, drop the bars and make the inside number positive Watch Out! -│6│ = - 6 because the negative is lurking outside the bars. ALGEBRA OPPOSITES Opposite of Multiplication is Division 25 * 4 = 100 100 ÷ 4 = 25 Opposite of Squaring is Square Rooting = 25 =5 Algebra Truths Adding Negative numbers - Think MONEY $$$$$$, You won’t miss it. No Equal Sign????? Simplify It!! Combine Like Terms. Simplifying is like cleaning up your room, put all the FROGS (Variables) together and all the SMILEY FACES (Numbers) together, then do the Math. Don’t Forget Numbers come with Signs! 4x -5 +6x +21 -8x 4x +6x -5 +21 -8x -------------------------------------------------------------------------------------------------------------------------------- 4x +6x -8x -5 +21 4x +6x -8x -5 +21 -------------------------------------------------------------------------------------------------------------------------------- 2x +16 2x +16 You can’t add FROGS & SMILEY FACES, so you are finished. Good Job! Adding or Subtracting **If signs are the same, add them and use the sign. 45 + 34 = 79, - 45 – 10 = -55 WATCH YOUR SIGNS Negative number digging the hole. Think Holes Negative is cooling off. Positive - dirt in the hole. Positive is warming up. **If signs are different, subtract and use sign of larger number -18 + 8 = -10, 60 – 20 = 40 Think temperature –Positive number, dollars in your pocket. Negative number, dollars borrowed Think money WATCH YOUR SIGNS Multiplying or Dividing **If signs are the same, answer is positive. 4 * 8 = 32, -63 / -7 = -9 **If signs are different, answer is negative. -6 * 7 = -42, -100 / 10 = -10 Distribution Principle Multiply everything in parenthesis by number next to parenthesis FIRST THING!! THINK BIG MOUTH SHOUTING “DO ME, DO ME FIRST!!” EXAMPLE : 3( 6x – 3) 3 (6x – 3) = 18x - 9 Distribution Principle Multiply everything in parenthesis by number next to parenthesis FIRST THING!! EXAMPLE : Get them off the bus So they can play football. 3 3( 6x – 3) 6x– 3 18x - 9 Equation Solving??? Example: 2(10x – 3) = 6x + 2 Get the Teams off the Bus Line up the Teams Penalty for off-sides – must change signs Huddle up Man on Man Defense Line of Scrimmage 2(10x – 3) = 6x + 2 20x – 6 = 6x + 8 20x – 6x = 8 + 6 14x = 14 14x = 14 14 14 X=1 Think Football – Letters Vs. Numbers ≤ , ›, or ‹, or ≥ If the equation has an Inequality sign, follow the steps for solving equations with = signs. Play Football Letters Vrs. Numbers Last Play of the Game If you have to MULTIPLY or DIVIDE by a negative number, Be sure to FLIP the Inequality. *****NOTICE***** UGLY NUMBERS WORK THE SAME AS PRETTY NUMBERS!!! UGLY Numbers Work the Same as PRETTY Numbers. If you can solve: 2X + 10 = 40 Play Football Letters Vrs. Numbers Then you can solve: 5x + 1.93 = 4.56 And you can solve: .5x + ⅝ = ⅓ Clue Words for writing equations from word problems Word Clue Word Sentence Algebraic Plus 1 plus 5 1+5 added to 6 is added to a number 6+x the sum of The sum of 5 and a number 5+x A number is increased by 10 x+10 15 is more than a number x+15 increasing by more than Clue Words for writing equations from word problems Word Clue Word Sentence Algebraic Minus 6 minus 5 6-5 subtracted from 7 subtracted from a number x-7 the difference of The difference of a number and 10 x-10 decreased by A number is decreased by 20 x-20 Less 5 less a number 5-x less than 6 less than a number x-6 Clue Words for writing equations from word problems Word Clue Word Sentence Algebraic Times 7 times a number 7*x = 7x Product Product of 8 and a number 8x Doubled A number doubled 2x Twice Twice a number 2*x = 2x Of (fractions and percents) 1/2 a number 1/2x 55% of a number 0.55x Clue Words for writing equations from word problems Word Clue Word Sentence Algebraic Quotient The quotient of a number and 7 x÷7 divided by 10 divided by a number 10÷x *The first number written before the clue word will be the numerator 1. Consistent – one or many solutions 2. Inconsistent – No solution 1. Independent – Only one solution 2. Dependent – Has infinitely many solutions. Slope Y – Int Graph Type of Systems # of Solutions Same Same Same line Consistent Dependent Infinitely many Same Different Parallel Inconsistent 0 Different Different/ Same Intersects Consistent Independent 1 Slope – Intercept Form y = mx + b Slope- directions Rise Run Y Intercept – where to start It’s a line address If the slope is a whole number, put it on a stick m = 2 To Graph: Example 1 slope is 2/1 Example 2 y = -3X+ 0 y = 2X + 1 y = -3X Starts at 1 Starts at 0 Rise/run = 2/1 rise/run = 3/-1 Directions are up 2, over 1 Directions are up 3, over -1 Thanks to http://www.mathsisfun.com/equation_of_line.html Linear Equations, Standard Form ax + by = c Solving for y, Just 3 easy steps Greenisms Math Terms 1.Move x term(Change sides, Change signs) 2.Give x side a hug 3.Divide by number next to the y Add/Subtract x term Parenthesis ( ) Coefficient Example: Solve for Y 2x – 7y = 12 Just 3 easy steps 1. -7y = 12 – 2x (Move x term (Change sides, Change signs) 2. -7y = (12-2x) (Give it a hug) 3. y = (12-2x) / -7 (Divide by number next to the y) Now you are ready to enter it into the calculator and graph it WATCH YOUR SIGNS!! Linear Equations, Standard Form ax + by = c Solving for y, It’s a Football Game Y VS Everybody Else Follow football rules Example: Solve for Y 2x – 7y = 12 Play Football Y vs everybody else Just 3 easy steps 1. -7y = 12 – 2x X is offside, Penalty change signs 2. -7y = (12-2x) Huddle up ( ) 3. y = (12-2x) / -7 Man on man defense Now you are ready to enter it into the calculator and graph it WATCH YOUR SIGNS!! y2 – y1 x2 – x1 Slopes are Negative Reciprocal (Flip & Change Sign) l lines or slope y = mx + b Find Equation of the Line: y = mx + b I need slope (m) & the y-intercept (b) To find m – Solve the equation for y and use m or use the y –y x – x formula 2 1 2 1 To find b - Plug x, y and m into the line equation and solve for b. MY ANSWER: y= x + (Just follow the Rules) Exponent Base x² Like Bases Exponent Multiply Add (x²a³g)(xa²g³) = (x³ a5 g ) Power up Multiply ( a5 g³ y) ³ = a15 g 9 y3 Divide Subtract s a³k a6 x4k 5 a3 x7k 4 = x³ Add No change 3a² + 5a² = 8a² a² + a5 = a² + a5 Subtract No change 10a² - 4a² = 6a² Example 4 x Example 5 4 3 x gk 2 2 4 ax k 1 5 = 5 x Switch Simplify g gx 2 3 7 2 5 4 3 axk k a xk 4 2 4 X X² X ∙ X = X² X X times X is X squared Quadratics Multiplying Binomials – Draw the Face (x + 8) (x – 6) Multiply Watch your Signs x² +8x -6x -48 Simplify (Combine Like Terms, eat the buggers) x² + 2x -48 Check This First The Math “F” Word “Factoring” 1. Is there a common factor (number or letter)? Yes Proceed with CGF NO. Example 8x²y + 4x³ - 12x Proceed to question 2 Factor each term 8x²y 2 * 2 * 2* x * x * y 4x³ 2*2*x*x*x 12x 2*2*3*x Circle common terms 8x²y 2 * 2 * 2* x * x * y 4x³ 2*2*x*x*x Multiply circled numbers That’s your Common Factor 12x 2*2*3*x Multiply leftovers, put in ( 4x ( 2xy + x² -3 ) ) 2. Perfect squares on end & 3 terms ? NO, proceed to question 3 x² - 10x + 25 YES, SPLIT IT NICE x² - 10x + 25 Put out baggies to hold the answer. (parenthesis) Split the first term nice (x - 5)(x - 5) Sign between is same as middle term Place in baggies ANSWER (x -5)² Split the second term nice. Place in baggies 3. Perfect squares on end & 2 terms? Yes, Same as question 2 Except the signs are +,- NO. proceed to question 4 Example x² - 64 Answer looks like this! ( x - 8 ) ( x + 8) 4. No perfect squares on end, 3 terms & starts with x²? YES, its quadratics in the morning NO, proceed to question 5 A M x² - 10x + 24 Multiply to +24 Add to -10 3,8 11 -3, -8 -11 -2, -12 -14 -6,-4 -10 Put in the parenthesis (x-6) (x-4) Found it All Done!! 5. Is there a number in front of the x² & does it have 3 terms Example 2x² + 7x + 3 Yes, Jail Break NO! Then it is Prime 1. Steal the “a” and give it to the last term (Multiply) 1 x² + 7x + 6 (2*3) (can’t factor) 2. Search and Seizure (quadratics in the morning)(See question 4) ( x+6 )( x+1) 3. Arrested and Caught - Divide last terms by “a” ( x + 6/2 ) ( x + 1/2 ) 4. Beat it Down - (reduce fractions) All Done!! ( x + 3 ) ( x + 1/2 ) (x + 3 ) ( 2 x + 1 ) 5. Parole (kick denominator to the front) The Math “F” Word “Factoring” Summary Check This First Is there a common factor (number or letter)? Is there a Square on each End?? Greatest Common Factor (GCF) Perfect Squares 1. 2. 3. 4. 5. 6. 7. Example Factor each term Circle common terms Multiply common terms Write it down Put out baggie for leftover Multiply leftovers for each term Put in baggie ( ) 8x²y + 4x³ - 12x 4x ( 2xy + x² - 3 ) 1. Put out Parenthesis 2. Split FIRST and LAST numbers nice 3. Put in Signs 3 Different Kinds A. x² - 16x + 64 (x– 8) (x – 8) B. x² + 18x + 81 (x + 9) (x + 9) C. x² - 36 (x + 6) (x – 6) The Math “F” Word “Factoring” Summary No perfect squares and 3 terms and starts with x²? Is there a number in front of the x² and does it have 3 terms? Quadratics in the Morning (AM) Jail Break 1. Make a factor tree 2. Multiply to last number, add to middle number 3. Put out baggies (parenthesis) 4. Split first term nice 5. Drop in factors from tree 1.Steal the “a” and give it to the last term (multiply) 2.Search and Seizure (Quad in AM) 3.Arrested and Caught (divide by “a”) 4.Beat Down (reduce fractions) 5.Parole (kick denominator to the front) 6.Check it out. (FACE it) Example 2x² + 7x + 3 Example x² + 12x + 32 (x + 8) (x + 4) 1x² + 7x + 6 2(x + 6) (x + 1) 3(x + 6/2) (x + 1/2) 4(x + 3) (x +1/2) 5(x + 3) (2x + 1) Systems of Equations To Solve by Graphing Solve with Graphing Calculator 1) Make an x, y Chart 1) 2) Select any x, solve for y x 2x – 4 y 0 2(0) – 4 -4 1 2(1) – 4 -2 3) Then graph the two points. 4) Do for both equations. 5) The answer is where they cross. 6) Be sure answer is in (x, y) form Solve each equation for y (3 easy steps) 2) Use y= Y= button and enter each equation 3) Use graph to eyeball answer Or 2) Use 2nd TABLE to find where y1 and y2 are equal 3) Be sure answer is in (x, y) form Systems of Equations Solve by Substitution (box & shove) 1) Solve one equation for x or y (change sides, change signs) 2) Box it 3) Rewrite other equation and shove box in 4) Solve for surviving letter 1) Distribute 2) Combine like terms 3) Solve 4) Send it back to box 5) Solve for other letter 6) Answer in (x,y) form 3y -2x = 11 y + 2x = 9 • y + 2x = 9 • y = 9 – 2x • 3(9 - 2x) – 2x = 11 • • • • • • • 27-6x – 2x = 11 27 – 8x = 11 - 8x = -16 x=2 y = 9 – 2 (2) y=5 (2, 5) Systems of Equations Solve by Elimination 2x – y = 9 3x + 4y = -14 1) Look for opposite signs • 2x – y = 9 3x + 4y = -14 2) Multiply to create opposites • 4(2x – y = 9) 3) Add old and new equation together 4) Solve for surviving letter • 8x – 4y = 36 • 3x + 4y = -14 • 11x = 22 • x=2 5) 6) Plug back into either equation (pick the easy one) Solve for other letter • 2(2) – y = 9 • -y = 5 • y = -5 7) Answer in (x,y) format • (2, -5)