ࡱ> #` ^bjbjmm 44T,,,,((($ $D8(HNIkh,BBBSZ|jjjjjjj$lhUoj(LSj,,BB k,&B(Bjj PR(xWB ЯQRFZk0IkRooxWo(xWjj"IkHNHNHN$HNHNHN<@ ,,,,,,  HYPERLINK \l "Solve1stepequ" Solving One-Step Equations HYPERLINK \l "Solvemultistepequ" Solving Multi-Step Equations HYPERLINK \l "eqwthdistributiveprop" Multi-Step Equations w/ Distributive Property HYPERLINK \l "variableseachside" Multi-step Equations with Variables on Each Side HYPERLINK \l "percentofchange" Percent of Change HYPERLINK \l "equationswthformula" Solving Equations with Formulas HYPERLINK \l "relations" Relations HYPERLINK \l "midpoint" Midpoint of a Line HYPERLINK \l "linearequations" Linear Equations HYPERLINK \l "howtographline" How To Graph a Line  HYPERLINK \l "findslope" Finding the Slope of a Line HYPERLINK \l "convertforms" Conversions Between Standard Form and Slope-Intercept Form HYPERLINK \l "parallelperpendicular" Parallel and Perpendicular Lines HYPERLINK \l "lineofbestfit" Line of Best Fit HYPERLINK \l "polynomials" Polynomial Operations HYPERLINK \l "powertopower" Power to a Power HYPERLINK \l "polybymono" Multiplying a Polynomial by a Monomial HYPERLINK \l "binomialstimesbinomials" binomials times binomials HYPERLINK \l "end" end  2/8/2012 Solving One-Step Equations 4 + ( = 10Algebra ( ( ( ( ( ( ( ( ( ( ( ( ( ( Concrete example 4 + x = 10Algebra with a variable To solve: 4 + x = 10 -4 - 4(= means same as) 0 + x = 6 x = 6This means that 6 is the only number that can replace x to make the equation true. ALWAYS SHOW YOUR WORK!! NO! 6 = X + 2 4 = XYES! 6 = X + 2 -2 -2ASK YOURSELF: What is happening to the variable? How can I get the variable by itself? 4 = X Examples: Addition/Subtraction Notice: In these examples, you are moving an entire quantity (the constant in these examples) to the other side of the equation. 1) 27 + n = 46 -27 -272) Remember that whatever you-5 + a = 21 +5 + 5 0 + n = 19 n = 19do on one side of the equation you have to do on the other! a = 26 Examples: Multiplication/Division Notice: In these examples, you are separating the quantity (the coefficient and the variable). 3)- 8n = -64 -8n = -64 -8 -8 n = 84) Remember that whatever you do on one side of the equation you have to do on the other! m = 12 7 7 * m = 12(7) 1 7 m = 84 5) 5b = 145 5b = 145 5 5 b = 296) -13 = m -5 (-13)(-5) = m * -5 -5 65 = m HYPERLINK \l "top" Return to Top 2/9/2012 Solving Multi-Step Equations Compare Equations EquationCommentsEquation 2 X + 3 = 15 X + 3 = 15 -3 -3( Total is the same ( Variable has a coefficient greater than 1 ( 2x + 3 = 15 2x + 3 = 15 -3 = -3 X = 12( One step Two steps ( 2x = 12 2x = 12 2 X = 6 Coefficient = the number in front of a variable Constant = a number. Example: in 2x + 3, 3 is the constant. Like Terms = quantities with the same variable (eg 2x, 0.5x, 644x, 342909835x, 1/2x, etc, are like terms because they all have an x). Numbers without a variable (constants) are also like terms! How to Solve Multi-Step Equations: Combine like terms on the left or right if you can. Get all your variables on one side and all your constants on the other side (addition/subtraction OR multiplication/division) Divide both sides by the coefficient of the variable. Examples: A) 7 y y = -1 1. Combine like terms.7 2y = -12. Get all variables on one side and all constants on the other. 7 2y = -1 -7 -7 -2y = -8 3. Divide both sides by the coefficient of the variable.-2y = -8 -2 -2 y = 4 B) 13 = 5 + 3b - 13 1. Combine like terms.13 = 3b - 82. Get all variables on one side and all constants on the other. 13 = 3b 8 +8 +8 21 = 3b 3. Divide both sides by the coefficient of the variable.21 = 3b 3 3 7 = b C) a 18 = 2 5z + 10 = 2 91. Combine like terms.2. Get all variables on one side and all constants on the other. a 18 = 2 5 +18 +18 a = 20 59(z + 10) = 2(9) 9 z + 10 = 18 z + 10 = 18 -10 -10 z = 83. Divide both sides by the coefficient of the variable. a * 5 = 20 * 5 5 a = 120  HYPERLINK \l "top" Return to Top 2/10/2012 Solving Multi-Step Equations That Include the Distributive Property Step 1: Distribute FirstExample: 2 = -2(n 4) 2 = -2(n) + (-2)(-4) 2 = -2n + 8 Step 2: Solve as you would any other multi-step equation - 8 = -2n 8 - 6 = -2n -6 = -2n -2 -2 3 = n -5(x + 3) = -45 -5x + (-15) = -45 +15 +15 -5x = -30 x = 6 This means that 6 is the only number that you can substitute to make this equation true. (  HYPERLINK \l "top" Return to Top 2/14/2012 Solving Multi-Step Equations with Variables on Each Side Step 1: Add like terms (there are none) Step 2: Distribute (none to do) Step 3: Variables on the leftExample: 6 x = 5x + 30 - 5x -5x 6 6x = 30Step 4: Numbers (constants) on the right -6 -6 -6x = 24 -6 -6 x = 4 Step 1: Add like terms (there are none) Step 2: Distribute (none to do) Step 3: Variables on the leftExample: 5x + 2 = 2x - 10 - 2x -2x 3x + 2 = - 10Step 4: Numbers (constants) on the right -2 -2 3x = -12 3 3 x = 4 Step 1: Add like terms Step 2: Distribute (none to do) Step 3: Variables on the left (if possible)Example: 5y 2y = 3y + 2 3y = 3y + 2 - 3y -3y 0 = 2Step 4: Numbers (constants) on the right0 does not equal 2. It is a false statement. This means that there are no real numbers that can be substituted for y in this equation. Step 1: Add like terms (there are none) Step 2: Distribute Step 3: Variables on the leftExample: 2y + 4 = 2(y + 2) 2y + 4 = 2(y) + 2(2) 2y + 4 = 2y + 4 ( -2y -2y 0 + 4 = 0 + 4 (Step 4: Numbers (constants) on the rightBoth sides of the equation are equal without a variable. Therefore, any real number can be substituted for y, and the equation will be true.  HYPERLINK \l "top" Return to Top 2/16/12 Percent of Change Types of INCREASES New = original(1 + r)Types of DECREASES New = original(1-r)* Tax (for example, on clothing purchases) * Interest (such as on a bank account) * inflation * commission (such as when a car salesman earns a certain % of what he sells)* sales * discounts * depreciation When solving a Percent of Change problem: EXAMPLE: original: 18 new: 10Step 1:WRITE OUT THE FORMULA!!!! If you dont it will be counted as wrong! New = original(1-r)Step 2:Organize your information.N =10 O = 18 r = ?Step 3:Plug your values into the equation.  10 = 18(1-r)Step 4:Solve the equation. 10 = 18 18r -18 = -18 - 8 = - 18r -18 -18 .4444 = r 44% = r  HYPERLINK \l "top" Return to Top 2/21/2012 Solving Equations with a Formula EXAMPLE: P = 2L + 2W If the length is 5 and the perimeter is 24, what is the value of W?Step 1:WRITE OUT THE FORMULA!!!! P = 2L + 2WStep 2:Organize your information.P = 24 L = 5 W = ?Step 3:Plug your values into the equation.  24 = 2(5) + 2WStep 4:Solve the equation. 24 = 10 + 2W -10 =-10 + 2W 14 = 2W 2 2 7 = W EXAMPLE: P = 2L + 2W If the width is 2.6 and the perimeter is 12.4, find L.Step 1:WRITE OUT THE FORMULA!!!! P = 2L + 2WStep 2:Organize your information.P = 12.4 L = ? W = 2.6Step 3:Plug your values into the equation.  12.4 = 2L + 2(2.6)Step 4:Solve the equation. 12.4 = 2L + 5.2 - 5.2 -5.2 7.2 = 2L 2 2 3.6 = L HYPERLINK \l "top" Return to the Top RELATIONS Domain = x values Range = y values f(x) = 2x2 + 7x 2 f(2a) = 2(2a)2 + 7(2a) 2 = 2(4a)2 + 14a 2 = 8a2 + 14a 2  HYPERLINK \l "top" Return to the top Midpoint of a Line The Midpoint of a Line is the middle, center, or halfway point of a line. Both segments of the line are equal distances.  Midpoint = m x + x , y + y 2 Example: Find the midpoint between (2, -4) and (-10, -3) 1. Identify your x and y coordinates: x y x y (2, -4) (-10, -3) 2. Put into the formula: 2 + (-10) , -4 + (-3) 2 2 -8 , -7 2 2 (-4, -7/2) or (-4, -3.5) or (-4, -3) Example: Using the midpoint formula, we will find the missing information: Given endpoint (-3, -7) Midpoint (-8, 4) Find endpoint (x, y) Solving by formula: x + x = xm , y + y = ym 2 2 2(-3 + x) = -8(2) , 2(-7 + y) = 4(2) 2 2 -3 + x = -16 , -7 + y = 8 +3 +3 +7 +7 x = - 13 , y = 15 (-13, 15)  HYPERLINK \l "top" Return to the Top 3/12/2012 Linear Equations * What makes an equation a linear equation? Linear = makes a line * What does linear look like? 1. No multiplication between variables together (in other words, no 2 variables are multiplied together) 2. No exponents greater than 1 EquationLinear or Non-LinearExplanationA.2x = 3y + 1LinearNo multiplication between variables; no exponents > 1B.4xy + 2y = 7Non-linearxyC.2x2 = 4y - 3Non-linearx2D.X 4y = 2 5 5 LinearNo multiplication between variables; no exponents >1 Linear equations can be written in standard form (Ax + By = C). X-intercept A point (x, y) where a line crosses the x-axis (x, 0) Y-intercept A point (x, y) where a line crosses the y-axis (0, y) How do you graph a line? Standard Form (Ax + By = C): use x and y intercepts Slope-Intercept Form (y = mx + b): use graphing calculator: y = 2nd graph Plot Graphing a line using Standard Form Plot the x-intercept Plot the y-intercept Write your equation twice. Cover (mark out) the x and its coefficient in one; cover (mark out) the y and its coefficient in the other Solve. Write your ordered pairs. Plot your points on a graph and draw your line through the points.Example: Graph 2x + 5y = 20  2x + 5y = 20 5y = 20 5 y = 4 x-intercept: (0, 4)  2x + 5y = 20 2x = 20 2 x = 10 y-intercept: (10, 0) Write your equation twice. Cover (mark out) the x and its coefficient in one; cover (mark out) the y and its coefficient in the other Solve. Write your ordered pairs. Plot your points on a graph and draw your line through the points.Example: Graph 3x + y = -1  3x + y = -1 y = -1 x-intercept: (0, -1)  3x + y = -1 3x = -1 3 x = -1/3 y-intercept: (-1/3, 0) Write your equation twice. Cover (mark out) the x and its coefficient in one; cover (mark out) the y and its coefficient in the other Solve. Write your ordered pairs. Plot your points on a graph and draw your line through the points.Example: Graph x y = -3  x - y = -3 -y = -3 You cant have a y, so multiply both sides by -1: -y(-1) = (-3)(-1) y = 3 x-intercept: (0, 3)  x - y = -3 x = -3 y-intercept: (-3, 0) Graphing a line using Slope-Intercept Form y = mx + b On your calculator, press: Y = 2nd graph (table) Pick out 3 or 4 points from the table that are easy to plot (ie X and Y are whole numbers) and plot them on a graph. Draw your line through those points.  HYPERLINK \l "top" Return to the top 3/14/12 Conversions Between Standard Form and Slope-Intercept Form Converting from Standard Form to Slope-Intercept Form Ax + By = C ( y = mx + b Example: Convert 4x + y = -2 to Slope-Intercept Form Write your original formula. Get y by itself. 4x + y = -2 4x -4x 0 y = -4x -2NOTE: You are not separating the coefficient from the variable (the x from the 4), so you are not dividing both sides by 4. You are moving the entire quantity of 4x to the other side of the equation, which is why you subtract. In other words, you are moving the variable, not isolating it. NOTE: Write the sentence so that it follows the form. Therefore, you write y = -4x 2, rather than y = -2 4x. Example: Convert 5x 3y = -6 to Slope-Intercept Form Write your original formula. Get y by itself (move x). 3. Solve so that y is positive and has a coefficient of 1 (isolate y and make it positive). 5x 3y = -6 5x -5x 0 -3y = -5x -6 -3 -3 -3 y = 5/3x + 2NOTE: You are not separating the coefficient from the variable (the x from the 5), so you are not dividing both sides by 5. You are moving the entire quantity of 5x, which is why you subtract. In other words, you are moving the variable, not isolating it. NOTE: Write the sentence so that it follows the form. Therefore, you write -3y = -5x 6, rather than -3y = -6 5x. NOTE: In this step you ARE separating the coefficient from the variable (the -3 from the y), so you DO divide. In other words, you are isolating the variable, not moving it. NOTE: Write the slope (m) as a FRACTION, not a decimal. Example: Convert 10x y = 6 to Slope-Intercept Form Write your original formula. Get y by itself (move x). 3. Solve so that y is positive and has a coefficient of 1 (isolate y and make it positive). 10x y = 6 10x -10x 0 (-1) -y = (-1)-10x + (-1)6 y = 10x + 6NOTE: You are not separating the coefficient from the variable (the x from the 5), so you are not dividing both sides by 10. You are moving the entire quantity of 10x, which is why you subtract. In other words, you are moving the variable, not isolating it. NOTE: Write the sentence so that it follows the form. Therefore, you write -y = -10x 6, rather than -y = -6 10x. Converting from Slope-Intercept Form to Standard Form y = mx + b ( Ax + By = C To be standard form: No fractions anywhere (includes decimals): all whole numbers Leading x cannot be negative Must be in simplest form 4x + 2y = 10 ( 2x + y = 5 2 2 2 Example 1: Change y = 3/2x 5 to Standard Form (Ax + By = C) 3/2x + y = (3/2x) (3/2x) 5 3/2x + y = -5 ( This is negative and a fraction, so it is not yet in standard form. (-2)(-3/2)x + (-2)y = -5(-2) -3x -2y = 10 Example 2: Change y = 1/5x 1 to Standard Form (Ax + By = C) 1/5x + y = (1/5x) (1/5x) 1 1/5x + y = -1 ( This is negative and a fraction, so it is not yet in standard form. (-5)(-1/5)x + (-5)y = -1(-5) x -5y = 10  HYPERLINK \l "top" Return to the top Finding the Slope of a Line Slope is rate of change It is the steepness of a line Represented by a fraction m = slope = rise = ( y = change in y run ( x change in x m = 4 = 4 1   Positive slope   Negative slope     Two ways to find slope: graph formula m = y y x x Example: Find the slope of the line going through the points (-4, 4) and (4, 6) m = y y x x 4 6 = -2 = 1 -4 4 -8 4 m = y y x x 6 - 4 = 2 = 1 4-(-4) 8 4 Example: Find the slope of the line going through the points (-4, -1) and (-2, -5) m = y y = -1 (-5) = 4 = -2 x x -4 (-2) -2  HYPERLINK \l "top" Return to the top 3/16/12 PARALLEL AND PERPENDICULAR LINES Parallel Lines (||) Do not intersect Have the same slopes Example: Write the slope-intercept form of an equation of a line parallel to y = 4x 2 and passing through the point (-2, 2) 1. Use the formula y y1 = m(x x1)y y1 = m(x x1)2. Organize your informationm = 4 x1 = -2 y1 = 23. Plug your values into your formula.y 2 = 4(x (-2))4. Solvey 2 = 4(x + 2) y 2 = 4x + 8 +2 +2 y = 4x + 10 Perpendicular Lines (() Intersect Slopes are 1) opposite (change the sign) 2) reciprocal (flip them) Example: Write the slope-intercept form of the ( line y = 1/2x + 1, crossing through point (4, 2). 1. Use the formula y y1 = m(x x1)y y1 = m(x x1)2. Convert your slope Opp flip m = ( - ( - 2/1 = -23. Organize your informationm = -2 x1 = 4 y1 = 24. Plug your values into your formula.y 2 = -2(x 4)5. Solvey 2 = -2x + 8 +2 +2 y = -2x + 10 Example: Write the slope-intercept form of the ( line 2x + 4y = 12, crossing through point (-1, 3). 1. Change the form of the original line from standard form to slope-intercept form. 2x + 4y = 12 - 2x -2x 4y = -2x 12 4 4 y = -1/2x - 31. Use the formula y y1 = m(x x1)y y1 = m(x x1)2. Convert your slope Opp flip m = - ( ( 2/1 = 23. Organize your informationm = 2 x1 = -1 y1 = 34. Plug your values into your formula.y 3 = 2(x (-1))5. Solvey 3 = 2x + 2 +3 +3 y = 2x + 5  HYPERLINK \l "top" Return to the top 3/19/12 Line of Best Fit Inside a SCATTERPLOT are data (ordered pairs). The more data you have, the better your line will be. Why do we need a line? TO PREDICT! (for example, it helps business owners predict profits, whether to expand or downsize, etc). Line of Best Fit Regression Equation Prediction Equation Best Fit Line(Equation of a line to predict. X (L1)Y (L2)Independent VariableDependent Variable ** See Best-Fit Line Handout** Types of Questions you can be asked with Line of Best Fit To find the equation To make a prediction (from a line) What does slope mean/represent? What does y-intercept represent? Example: The cab driver charges a $5 flat fee and $.25 per mile. y = .25x + 5 y represents the total bill .25 represents the charge or RATE x represents per mile 5 represents the flat fee or starting point; the total bill starting at 0 miles  HYPERLINK \l "top" Return to the Top 4/9/12 Polynomial Operations Monomial: one term A number, a variable, or the product of 1 or more variables Examples: 5, a, 6x, 5x2yz Binomial: two terms Addition or subtraction of two monomials Examples: 5-5x; 2x2+3 Trinomial: three terms Addition or subtraction of three monomials Example: 6+7x2-3x Adding Polynomials: (4a - 5) + (3a + 6) Combine like terms/regroup (4a - 5) + (3a + 6) 4a + 3a - 5 + 6 7a + 1 (6xy + 2y + 6x) + (4xy x) 10xy + 2y + 5x Subtracting Polynomials: 1)(3a 5) (5a + 1)DISTRIBUTE the -1 on the 2nd monomial(3a 5) + (- 5a 1)Combine like terms- 2a - 6 2)(9xy + y 2x) (6xy 2x)DISTRIBUTE the -1 on the 2nd monomial(9xy + y 2x) +(- 6xy + 2x)Combine like terms3xy + y + 0 3xy + y  HYPERLINK \l "top" Return to top 4/10/12 Multiplying Monomials Multiply coefficient (whole number) Add exponents What does y3 mean? y * y * y What does (x2)(x4) mean? x * x |* x * x * x * x ( x6 x2+4 = x6 EXAMPLES: (x16) (x163) = x16+163 = x179 (4x3y2) (-2xy) (4) (-2) x3+1 y2+1 -8 x4 y3 3) (-5xy) (4x2) (y4) (-5) (4) (1) x3+1 y2+1 4) (-3j2k4) (2jk6) (-3) (2) j2+1 k4+6 -6j3k10 Power to a Power (x2)3 = x2 * x2 * x2 = x2+2+2 = x6 x x x x x x = x6 x2*3 = x6 RULE: Multiply exponents. EXAMPLES: (xy3)4 = xy3 * xy3 * xy3 * xy3 = x4 x12 = x1*4 y3*4 = x4 x12 4/11/12 Multiplying a Polynomial by a Monomial RULE: Distribute! EXAMPLES: 1) x(5x + x2) x(5x) + x(x2) 5x2 + x3 2) -2xy(2xy + 4 x2) (-2xy) (2y) + (-2xy) (4 x2) -4x y2 + -8 x3y 3) 2x2y2(3xy + 2y + 5x) 6x3y3 + 4x2y3 + 10x3y2  HYPERLINK \l "top" Return to top 4/12/12 Binomials Times Binomials There are 3 techniques you can use for multiplying polynomials: Distributive Property FOIL Method The Box Method EXAMPLE OF DISTRIBUTIVE PROPERTY OR FOIL METHOD: (2x + 3) (5x + 8) First: (2x)(5x) = 10x2 Outer: (2x)(8) = 16x Inner: (3)(5x) = 15x Last: (3)(8) = 24 Combine like terms: 10x2 + 31x + 24 EXAMPLES OF BOX METHOD: (3x 5) (5x + 2) 3x-55x15 x2-25x15 x2 25x + 6x 1026x-1015 x2 19x 10  (2x 5) (x2 5x + 4) x2-5x42x3 - 10x2 - 5x2 + 8x + 25x - 202x2 x3-10 x28x2x3 - 15x2 + 33x - 20-5-5x225x-20 4/13/12 Dividing Monomials When dividing monomials, subtract the exponents 1. b5 = b * b * b * b * b = b5-2 = b3 b2 b * b end     H. Bullard & L. 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