Signed Numbers: A signed number is positive + (or negative - ) from 0. Addition: To add signed numbers that are alike in sign, add and keep the same sign. Add: -5 +7 + -3 + +6 -- --- -8 +13 :RA If the signs are unlike, subtract and use the sign of the largest absolute value. Add: -8 + +3 -- -5 :RA Subtraction: Change the sign of the bottom number and follow the same rules as in addition. Subtract: +7 - -3 --- +10 Subtract: (+3) - (+5) = (+3) + (-5) = -2 :RA Multiplication: To multiply two numbers with the same sign, the answer is positive (+). Multiply: (+3)(+7) = +21 (-4)(-3) = +12 To multiply two numbers with different signs, the answer is negative (-). Multiply: (-5)(+4) = -20 (+4)(-6) = -24 :RA Division: The same rules apply as in multiplication. Divide: +6 -- = +3 +2 -8 -- = -2 +4 :RA Linear Equations: Order Of Operations Rules: 1. Remove fractions and decimals by multiplication. 2. Remove parenthesis by multiplication. 3. Combine similar terms on each side of the equation. 4. Collect unknowns on one side of the equation and constants on the other side. Note: When a term crosses the equal sign it changes its sign. :RA 5. Divide both sides by the coefficient. Note: Use the same number as the coefficient, not the opposite sign. Example: 7y + 3 = 3y + 19 4y + 3 = 19 4y = 16 y = 4 Ans. :RA Example: 3/4 y + 2 = 2/3 y + 3 Multiply by 12: 9y + 24 = 8y + 36 y + 24 = 36 y = 12 Ans. Example: 5/9 x + 2 = 7/2 x - 3 Multiply by 18: 10x + 36 = 63x - 54 10x = 63x - 90 -53x = -90 x = 1.7 Ans. :RA Literal Equations: 1. Have more than one variable. 2. Follow same rules as numerical equations. Example: ax + b = c + d Solve for x: ax c + d - b -- = --------- a a c + d - b x = --------- Ans. a :RA Systems Of Linear Equations 1. Eliminate one unknown by linear combination. Example: Solve for x: 4x - 2y = 2 5x - 2y = 14 Multiply by -1: -1(5x - 2y = 14) 4x - 2y = 2 -5x + 2y = -14 -------------- -x = -12 x = 12 Ans. :RA Quadratic Equations: 1. Generally have 2 roots. 2. Get all equal to 0. 3. Solve by factoring. 2 Example: x + 7x + 12 = 0 (x + 4)(x + 3) = 0 x = -4 or -3 Ans. :RA 2 Example: x - 16 = 0 2 x = 16 x = + or - 4 Ans. :RA 4. If you cannot solve by factoring, use the quadratic formula. _______ / 2 x = -b +/- \/ b -4ac ----------------- 2a :RA 2 Example: x - 2x - 1 = 0 ___________ 2 +/- \/4 -4(1)(-1) ------------------- 2 _ 2 +/- \/8 = --------- 2 _ = +/- \/2 Ans. :RA Radical Equations When solving equations with radicals, put the radical alone on one side, then square both sides to remove the radical. You must check your answer(s). In some cases one or more answers must be rejected. _ Example: \/y = -6 y = 36 Ans. _____ Example: \/y + 5 = 7 y + 5 = 49 y = 44 Ans. :RA Example: _ _ _ 4\/2 + 3\/2 = 7\/2 Ans. Note: Only like radicals can be added; for example: _ _ \/2 cannot be added to 2\/3. :RA Note: Radicals are simplified by removing any perfect square factors. ___ ___ Example: \/578 + \/450 = ? ___ ___ \/289x2 + \/225x2 = ? _ _ _ 17\/2 + 15\/2 = 32\/2 Ans. :RA In simplifying radicals that contain a sum or difference in the radical sign; first add or subtract and then take the square root. :RA _________ Example: / 2 2 / y + y / -- -- \/ 9 16 ___________ / 2 2 / 16y + 9y / ----------- = \/ 144 ____ / 2 / 25y 5y / ---- = -- Ans. \/ 144 12 :RA Reducing Algebraic Fractions: 1. To reduce algebraic fractions, divide the numerator and denominator by the same factor. Do NOT cancel terms. 2 2y - 8y Example: Reduce: --------- 4 3 4y -16y 2y (y-4) 1 -------- = --- Ans. 3 2 4y (y-4) 2y :RA 2 5 Example: Find the sum: - + - C y (The lowest common denominator is Cy): 2 y 5 C - x - + - x - C y y C 2y 5C = -- + -- Cy Cy 2y + 5C = ------- Ans. Cy :RA Example: Multiply: Find the product: 3 3 a b -- x -- 2 2 b a 3 a b = -- x -- 2 1 b a b = - x - = ab Ans. 1 1 :RA Complex algebraic fractions are simplified just as in arithmetic. Multiply each term of the complex fraction by the common denominator. 1 1 - + - a b Example: ----- ab b + a ----- Ans. 2 2 a b Multiply each term by ab: Note: DO NOT CANCEL TERMS. :ET :ET
