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ALGEBRA I NOTES H. Bullard & L. Mills Solving One-Step Equations Multi-Step Equations w/ Distributive Property Percent of Change Relations Midpoint of a Line How To Graph a Line Finding the Slope of a Line Parallel and Perpendicular Lines Polynomial Operations Multiplying a Polynomial by a Monomial end 2/8/2012 Solving One-Step Equations 4 + = 10 + ? = Solving Multi-Step Equations Multi-step Equations with Variables on Each Side Solving Equations with Formulas Linear Equations Conversions Between Standard Form and SlopeIntercept Form Line of Best Fit Power to a Power binomials times binomials Algebra Concrete example 4 + x = 10 To solve: Spring 2012 Algebra with a variable 4 + x = 10 -4 -4 0+x= 6 x= 6 (“=” means “same as”) This means that 6 is the only number that can replace x to make the equation true. ALWAYS SHOW YOUR WORK!! NO! YES! 6=X+2 6=X+2 4=X -2 -2 4=X ASK YOURSELF: What is happening to the variable? How can I get the variable by itself? 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 2) -5 + a = 21 -27 -27 Remember that whatever you +5 +5 0 + n = 19 do on one side of the equation you have to do on the other! n = 19 a = 26 Examples: Multiplication/Division Notice: In these examples, you are separating the quantity (the coefficient and the variable). 3) - 8n = -64 4) m = 12 Remember that whatever you 7 -8n = -64 do on one side of the equation -8 -8 you have to do on the other! 7 * m = 12(7) 1 7 Page 1 H. Bullard & L. Mills ALGEBRA I NOTES Spring 2012 n=8 m = 84 5) 5b = 145 6) 5b = 145 5 5 -13 = m -5 (-13)(-5) = m * -5 -5 65 = m b = 29 Return to Top 2/9/2012 Solving Multi-Step Equations Compare Equations Equation X + 3 = 15 X + 3 = 15 -3 -3 X = 12 Comments Total is the same Variable has a coefficient greater than 1 One step Equation 2 2x + 3 = 15 2x + 3 = 15 -3 = -3 2x = 12 2x = 12 2 2 Two steps 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: 1. Combine like terms on the left or right if you can. 2. Get all your variables on one side and all your constants on the other side (addition/subtraction OR multiplication/division) 3. Divide both sides by the coefficient of the variable. Examples: A) 7 – y – y = -1 1. Combine like terms. 2. Get all variables on one side and all constants on the other. 7 – 2y = -1 7 – 2y = -1 -7 -7 -2y = -8 3. Divide both sides by the coefficient of the variable. -2y = -8 -2 -2 Page 2 H. Bullard & L. Mills ALGEBRA I NOTES Spring 2012 y=4 B) 13 = 5 + 3b - 13 1. Combine like terms. 2. Get all variables on one side and all constants on the other. 13 = 3b - 8 13 = 3b – 8 +8 +8 21 = 3b 3. Divide both sides by the coefficient of the variable. 21 = 3b 3 3 7=b C) 1. Combine like terms. 2. Get all variables on one side and all constants on the other. a – 18 = 2 5 z + 10 = 2 9 a – 18 = 2 5 +18 +18 a = 20 5 9(z + 10) = 2(9) 9 z + 10 = 18 z + 10 = 18 -10 -10 z =8 3. Divide both sides by the coefficient of the variable. a * 5 = 20 * 5 5 a = 120 Return to Top 2/10/2012 Solving Multi-Step Equations That Include the Distributive Property Step 1: Distribute First Step 2: Solve as you would any other multi-step equation Example: 2 = -2(n – 4) 2 = -2(n) + (-2)(-4) 2 = -2n + 8 - 8 = -2n – 8 - 6 = -2n -6 = -2n -2 -2 3=n Page 3 ALGEBRA I NOTES H. Bullard & L. Mills -5(x + 3) = -45 -5x + (-15) = -45 +15 +15 -5x = -30 x=6 Spring 2012 This means that 6 is the only number that you can substitute to make this equation true. 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 left Step 4: Numbers (constants) on the right Example: 6 – x = 5x + 30 - 5x -5x 6 – 6x = 30 -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 left Step 4: Numbers (constants) on the right Example: 5x + 2 = 2x - 10 - 2x -2x 3x + 2 = - 10 -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) Step 4: Numbers (constants) on the right Step 1: Add like terms (there are none) Step 2: Distribute Step 3: Variables on the left Step 4: Numbers (constants) on the right Example: 5y – 2y = 3y + 2 3y = 3y + 2 - 3y -3y 0=2 0 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. Example: 2y + 4 = 2(y + 2) 2y + 4 = 2(y) + 2(2) 2y + 4 = 2y + 4 -2y -2y 0+4=0+4 Both sides of the equation are equal without a variable. Therefore, any real number can be substituted for y, and the equation will be true. Page 4 H. Bullard & L. Mills ALGEBRA I NOTES Spring 2012 Return to Top 2/16/12 Percent of Change Types of INCREASES 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) Types of DECREASES New = original(1-r) * sales * discounts * depreciation When solving a Percent of Change problem: Step 1: WRITE OUT THE FORMULA!!!! If you don’t it will be counted as wrong! Step 2: Organize your information. Step 3: Plug your values into the equation. Step 4: Solve the equation. EXAMPLE: original: 18 New = original(1-r) new: 10 N =10 O = 18 r=? 10 = 18(1-r) 10 = 18 – 18r -18 = -18 - 8 = - 18r -18 -18 .4444 = r 44% = r Return to Top 2/21/2012 Solving Equations with a Formula Step 1: WRITE OUT THE FORMULA!!!! Step 2: Organize your information. Step 3: Plug your values into the equation. Step 4: Solve the equation. EXAMPLE: P = 2L + 2W If the length is 5 and the perimeter is 24, what is the value of W? P = 2L + 2W P = 24 L=5 W=? 24 = 2(5) + 2W 24 = 10 + 2W -10 =-10 + 2W 14 = 2W 2 2 7=W Page 5 ALGEBRA I NOTES H. Bullard & L. Mills Step 1: WRITE OUT THE FORMULA!!!! Step 2: Organize your information. Step 3: Plug your values into the equation. Step 4: Solve the equation. Spring 2012 EXAMPLE: P = 2L + 2W If the width is 2.6 and the perimeter is 12.4, find L. P = 2L + 2W P = 12.4 L=? W = 2.6 12.4 = 2L + 2(2.6) 12.4 = 2L + 5.2 - 5.2 -5.2 7.2 = 2L 2 2 3.6 = L 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 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 2 ( ) Example: Find the midpoint between (2, -4) and (-10, -3) 1. Identify your x and y coordinates: 2. Put into the formula: 2 + (-10) , 2 x y (2, -4) x y (-10, -3) -4 + (-3) 2 -8 , -7 2 2 (-4, -7/2) or (-4, -3.5) or (-4, -3½) Page 6 H. Bullard & L. Mills ALGEBRA I NOTES Spring 2012 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) 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 A. B. C. D. Equation 2x = 3y + 1 Linear or Non-Linear Linear 4xy + 2y = 7 2x2 = 4y - 3 X – 4y = 2 5 5 Non-linear Non-linear Linear 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? 1. Standard Form (Ax + By = C): use x and y intercepts 2. Slope-Intercept Form (y = mx + b): use graphing calculator: y= 2nd graph Plot Page 7 Explanation No multiplication between variables; no exponents > 1 xy x2 No multiplication between variables; no exponents >1 H. Bullard & L. Mills ALGEBRA I NOTES Spring 2012 1. Graphing a line using Standard Form Plot the x-intercept Plot the y-intercept find them by using the “Thumb Rule” Example: Graph 2x + 5y = 20 1. Write your equation twice. 2. Cover (mark out) the x and its coefficient in one; cover (mark out) the y and its coefficient in the other 3. Solve. 4. Write your ordered pairs. 5. Plot your points on a graph and draw your line through the points. 2x X + 5y = 20 This means x is 0, so this ordered pair will be (0, ) 5y = 20 5 5 2x + 5y X = 20 This means y is 0, so this ordered pair will be ( , 0 ) 2x = 20 2 2 y=4 x-intercept: (0, 4) x = 10 y-intercept: (10, 0) Example: Graph 3x + y = -1 1. Write your equation twice. 2. Cover (mark out) the x and its coefficient in one; cover (mark out) the y and its coefficient in the other 3. Solve. 4. Write your ordered pairs. 5. Plot your points on a graph and draw your line through the points. 3x X + y = -1 This means x is 0, so this ordered pair will be (0, ) y = -1 3x +Xy = -1 This means y is 0, so this ordered pair will be ( , 0 ) 3x = -1 3 3 x-intercept: (0, -1) x = -1/3 y-intercept: (-1/3, 0) Example: Graph x – y = -3 1. Write your equation twice. 2. Cover (mark out) the x and its coefficient in one; cover (mark out) the y and its coefficient in the other 3. Solve. 4. Write your ordered pairs. 5. Plot your points on a graph and draw your line through the points. x - y = -3 X This means x is 0, so this ordered pair will be (0, ) -y = -3 You can’t have a –y, so multiply both sides by -1: -y(-1) = (-3)(-1) y=3 x-intercept: (0, 3) Page 8 x -X y = -3 This means y is 0, so this ordered pair will be ( , 0 ) x = -3 y-intercept: (-3, 0) H. Bullard & L. Mills ALGEBRA I NOTES Spring 2012 2. 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. Return to the top 3/14/12 Conversions Between Standard Form and Slope-Intercept Form 1. Converting from Standard Form to Slope-Intercept Form Ax + By = C y = mx + b Example: Convert 4x + y = -2 to Slope-Intercept Form 1. Write your original formula. 4x + y = -2 – 4x 2. Get y by itself. -4x 0 y = -4x -2 Example: Convert 5x – 3y = -6 to Slope-Intercept Form 1. Write your original formula. 5x – 3y = -6 – 5x 2. Get y by itself (move x). -5x 0 -3y = -5x -6 3. Solve so that y is positive and -3 -3 -3 has a coefficient of 1 (isolate y and make it positive). y = 5/3x + 2 NOTE: 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. NOTE: 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 2. Write your original 10x – y = 6 – 10x formula. -10x 2. Get y by itself (move x). 0 (-1) -y = (-1)-10x + (-1)6 3. Solve so that y is positive and has a coefficient of 1 (isolate y and make it positive). y = 10x + 6 Page 9 NOTE: 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. H. Bullard & L. Mills ALGEBRA I NOTES Spring 2012 2. Converting from Slope-Intercept Form to Standard Form y = mx + b Ax + By = C To be standard form: 1. No fractions anywhere (includes decimals): all whole numbers 2. Leading x cannot be negative 3. 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 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 Zero slope (m = 0) negative positive Undefined (no slope) Page 10 H. Bullard & L. Mills ALGEBRA I NOTES Spring 2012 Two ways to find slope: 1. graph 2. 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 m=y–y x–x x–x 4 – 6 = -2 = 1 -4 – 4 -8 4 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 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 information m=4 x1 = -2 y1 = 2 3. Plug your values into your formula. y – 2 = 4(x – (-2)) 4. Solve y – 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) Page 11 H. Bullard & L. Mills ALGEBRA I NOTES Spring 2012 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 = ½ - ½ - 2/1 = -2 3. Organize your information m = -2 x1 = 4 y1 = 2 4. Plug your values into your formula. y – 2 = -2(x – 4) 5. Solve y – 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 2x + 4y = 12 - 2x from standard form to slope-intercept -2x form. 4y = -2x – 12 4 4 1. Use the formula y – y1 = m(x – x1) 2. Convert your slope 3. Organize your information 4. Plug your values into your formula. 5. Solve y = -1/2x - 3 y – y1 = m(x – x1) Opp flip m = -½ ½ 2/1 = 2 m=2 x1 = -1 y1 = 3 y – 3 = 2(x – (-1)) y – 3 = 2x + 2 +3 +3 y = 2x + 5 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). Page 12 ALGEBRA I NOTES H. Bullard & L. Mills Line of Best Fit Regression Equation Prediction Equation Best Fit Line Spring 2012 Equation of a line to predict. X (L1) Y (L2) Independent Variable Dependent Variable ** See “Best-Fit Line” Handout** Types of Questions you can be asked with Line of Best Fit 1. To find the equation 2. To make a prediction (from a line) 3. What does slope mean/represent? 4. 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 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: 1) (4a - 5) + (3a + 6) Combine like terms/regroup (4a - 5) + (3a + 6) 4a + 3a - 5 + 6 7a + 1 Page 13 H. Bullard & L. Mills ALGEBRA I NOTES 2) (6xy + 2y + 6x) + (4xy – x) 10xy + 2y + 5x Subtracting Polynomials: 1) DISTRIBUTE the -1 on the 2nd monomial Combine like terms (3a – 5) – (5a + 1) (3a – 5) + (- 5a – 1) - 2a - 6 2) DISTRIBUTE the -1 on the 2nd monomial Combine like terms (9xy + y – 2x) – (6xy – 2x) (9xy + y – 2x) +(- 6xy + 2x) 3xy + y + 0 3xy + y Return to top 4/10/12 Multiplying Monomials 1. Multiply coefficient (whole number) 2. 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: 1) (x16) (x163) = x16+163 = x179 2) (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 xx xx xx = x6 2*3 6 x = x RULE: Multiply exponents. EXAMPLES: 1) (xy3)4 = xy3 * xy3 * xy3 * xy3 = x4 x12 = x1*4 y3*4 = x4 x12 Page 14 Spring 2012 ALGEBRA I NOTES H. Bullard & L. Mills 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 Return to top 4/12/12 Binomials Times Binomials There are 3 techniques you can use for multiplying polynomials: 1. Distributive Property 2. FOIL Method 3. 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 -5 2 5x 15 x -25x 15 x2 – 25x + 6x – 10 2 6x -10 15 x2 – 19x – 10 (2x – 5) (x2 – 5x + 4) x2 -5x 2x 2 x3 -10 x2 -5 -5x2 25x 4 8x -20 2x3 - 10x2 - 5x2 + 8x + 25x - 20 2x3 - 15x2 + 33x - 20 Page 15 Spring 2012 H. Bullard & L. Mills ALGEBRA I NOTES 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 Page 16 Spring 2012