Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Location arithmetic wikipedia , lookup
List of important publications in mathematics wikipedia , lookup
Line (geometry) wikipedia , lookup
Recurrence relation wikipedia , lookup
System of polynomial equations wikipedia , lookup
System of linear equations wikipedia , lookup
Partial differential equation wikipedia , lookup
History of algebra wikipedia , lookup
Shady Side Academy Middle School Review Packet for students entering Geometry * Some work taken from Geometry, by Jurgensen, Brown & Jurgensen, Houghton Mifflin Company, 1997 DIRECTIONS: Read through the information and study the given examples. Complete all boxed exercises on notebook paper using proper form. SECTION I: LINEAR EQUATIONS & PROPERTIES OF EQUALITY A. Solving Linear Equations To solve a linear equation, use the operations that are the opposite of the ones in the equation (inverse operations). Examples 1. Solve to find the value of the variable. Show proper algebraic transformations. x + 7 = 12 2. 3m + (m – 2) = 10 3. 4t + 23 = 9t – 7 x + 7 – 7 = 12 – 7 3m + m + (–2) = 10 4t + 23 + 7 = 9t – 7 + 7 x=5 4m + (–2) = 10 4t + 30 = 9t 4m + (–2) + 2 = 10 + 2 4t + (–4t) + 30 = 9t + (–4t) 4m = 12 30 = 5t 4m 12 4 4 30 5t 5 5 m=3 6=t Exercises – Find the value of the variable in the given equation. 1. (n – 20) + 5n = 28 3 a =9 4 2. 3. 2(h + 5) = 3(h – 2) B. Properties of Equality for all real numbers Addition Property If a = b and c = d, than a + c = b + d Multiplication Property If a = b, then ca = cb Substitution Property If a = b, then either a or b may be substituted for the other in any equation. Reflexive Property a=a Symmetric Property If a = b, then b = a Transitive Property If a = b and b = c, then a = c Exercises – Name the property that justifies the algebraic transformation shown. 1. 4b – 5 = –2 2. 15f + 7 = 12 – 20 f 4b = 3____________ 35f + 7 = 12____________ b = 0.75__________ 35f = 5 ________________ f= 1 _______________ 7 3. If k 5 = 10, then k = 50 _____________________ C. Systems of Linear Equations Example 1 Solve the given system by the Substitution Method: y = 5 – 2x and 5x – 6y = 21 1. Substitute 5 – 2x for y in the second equation, 2. Now substitute 3 for x then solve for x: in the other equation: 5x – 6(5 – 2x) = 21 5(3) – 6y = 21 5x 6(5) – 6(–2x) = 21 15 – 6y = 21 5x –30 + 12x = 21 –6y = 6 17x – 30 = 21 y = –1 S = (3, –1) 17x = 51 x=3 Example 2 Solve the given system by the Linear Combination Method: 6x – 2y = 26 6x – 2y = 26 4x + y = 22 Multiply by 2 8x + 2y = 44 14x = 70 x=5 Substitute 5 for x in one of the original equations: 4(5) + y = 22 20 + y = 22 y=2 S = (5, 2) Exercises – Solve each system of linear equations. 1. y = 3x 5x + y = 24 2. 3x + y = 19 2x – 5y = -10 3. 5x + 2y = 19 3x – 4y = 1 4. 3x + 2y = 71 2x – y = –4 SECTION II: OPERATIONS WITH NUMBERS To show work in proper form, you must do the following: 1. Write the given expression. 2. Complete the operations within any symbols of inclusion – parentheses, brackets, vinculums (aka fraction bars). Do the innermost operation and write the result. Use an = sign to connect the new expression to the original one. 3. Keep doing the operations until you reduce the expression to a single number. Use = signs to connect each expression to the one before (see examples below). Example 1 Evaluate 24 8 2 2 24 8 2 2 = 24 4 2 = 24 2 = 48 Example 2 Evaluate 24 8 2 2 24 8 2 2 = 24 8 2 2 = 192 2 2 = 96 2 = 94 Example 3 Evaluate 68 4 28 68 48 4= 4 28 10 4 108 + 4 = 4 54 4 = 8 54 Exercises – Evaluate the following expressions showing work in proper form . 1. 2. 14 6 9 2 14 6 9 2 4. 800 50 2 7 5. 19 25 7 54 3. 800 50 2 7 6. 28 8 6 57 SECTION III – VARIABLES AND SUBSTITUTION A variable is a letter that represents a number. To evaluate an expression containing a variable, your must be told what value to use for the variable, then you substitute that number for the variable and evaluate the expression as before. Example 1 Evaluate the expression by substituting the given values of the variable. x +7, if : a) x is 8; b) x is 93. a) x + 7 = 8 + 7 b) x + 7 = 93 + 7 = 15 = 100 Example 2 Evaluate the expression by substituting the given values of the variable. a) 40 40 = x 5 b) =8 40 40 = x 80 40 40 = 80 40 1 = 2 40 , if: x a) x is 5; b) x is 80; c) x is 60. 40 40 = x 60 40 20 = 60 20 2 = 3 c) Exercises – Evaluate the expressions by substituting in the given values of the variable. 1. 72 n 5 , if: 2. a) n is 6; b) n is 9; c) n is 0. y 31 4 , if: 3. 7x, if: a) y is 80; b) y is 57; c) y is 31. 4. a) x is 8; b) x is 11. k7 , if: k 3 a) k is 5; b) k is 3; c) k is -7. SECTION IV – POWERS AND EXPONENTS In the expression 45, the 4 is called the base, the 5 is called the exponent, and the whole expression is called a power. To evaluate a power, you simply do the multiplication. Example 1 Evaluate 53 3 Evaluate 4 Example 2 5 = 555 3 3 = 2 4 4 = 125 = 2 2 3 2 9 16 ***IT IS IMPORTANT TO KNOW THE DIFFERENCE BETWEEN 3 AND 32 !!! 2 3 2 means multiply –3 and –3 together to get +9. 32 means “the opposite of 32 ” which is –9. 3 2 = 3 3 = +9 32 = 3 = 3 3 = –9 2 SECTION V – ORDER OF OPERATIONS In order to evaluate a given expression, you must perform the given mathematical operations following the order of operations. The order is: 1. Complete the operations within any symbols of inclusion – parentheses, brackets, vinculums (aka ‘fraction bars’). 2. Evaluate any exponents. 3. Divide and/or multiply, in order, from left to right. 4. Subtract and/or add, in order, from left to right. Example 1 Evaluate 60 7 5 6 2 24 60 7 5 6 2 24 = 60 7 5 3 24 Inside the ( ), divide before adding. Example 2 = 60 7 8 24 Do what is inside the ( ) first. = 60 7 8 16 = 60 56 16 = 4 16 = 20 Evaluate the exponent. Multiply before adding or subtracting. Add & subtract in order from left to right. Answer Evaluate 3x 2 5x if x is 4. 3x 2 5x = 3 4 5 4 = 3 16 5 4 = 48 20 = 28 2 Substitute 4 for x in both places. Use ( ) with the exponent. Raise to powers before multiplying. Multiply before adding or subtracting. Answer Exercises – Evaluate the given expressions using order of operations. 1. a) 5 2 4 3 b) c) 4. 2. 5 2 4 3 5 2 4 3 30 10 5 3 a) 7 32 3. b) 7 32 c) 5. 7 3 8 5 43 x 2 2x 1 , if : a) x is 6; 2 b) x is 4. 6. 74 4 3 9 4 52 SECTION VI – QUADRATIC EQUATIONS All quadratic equations in standard form can be solved using the Quadratic Formula. In other words, if ax2 + bx + c = 0, with a 0, then x b b 2 4ac . 2a If the quadratic equation can be factored, it can be solved quickly that way. Example 1 Example 2 3x2 + 14x + 8 = 0 3x2 + 14x + 8 = 0 (3x + 2)(x + 4) = 0 x b b 2 4ac 2a 3x + 2 = 0 or x + 4 = 0 x 14 142 4(3)(8) 2(3) x 14 196 96 14 10 6 6 x 2 or x = -4 3 x a = 3, b = 14, c = 8 2 or x = -4 3 Exercises – Solve each quadratic equation. Factor where you can if you like. 1. 2m2 + m = 0 2. y2 + 8y + 12 = 0 3. 3x2 + 3x = 4 4. x(x + 5) = 14 SECTION VII – RATIONAL ALGEBRAIC EXPRESSIONS The rules for fractions apply to all rational algebraic expressions: 1. a c a+c + = ,b0 b b b Example 1 8w =2w 4 2. a c ac • = , b 0, d 0 b d bd Example 2 5t - 10 5(t - 2) 15 15 t-2 3 3. a c a d ÷ = • , b 0, c 0, d 0 b d b c Example 3 x+6 x+6 2 36 - x (6 - x)(6+ x) 1 = 6-x Exercises – Simplify the rational algebraic expressions below. 1. -18r 3 t = 12rt 2. 33ab - 22b 11b 5a + 5b = a 2 - b2 3. 4. 3x 2 - 6x - 24 = 3x 2 + 2x - 8 SECTION VIII – RADICAL ALGEBRAIC EXPRESSIONS The symbol always indicates the positive square root of a number. The radical Example 1 Example 2 16 = 3 56 = 4 14 16 3 4 3 3 3 4 3 3 4 14 2 14 64 can be simplified. Example 3 (3 7)2 = (3 7)(3 7) 3 3 7 7 9 7 63 Exercises – Simplify the rational algebraic expressions below. 1. 98 2. 250 48 3. (9 2) 2 4. 300 30