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Unformatted text preview: Solving Equations SFDR Algebra I Say Thanks to the Authors Click (No sign in required) To access a customizable version of this book, as well as other interactive content, visit CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-source, collaborative, and web-based compilation model, CK-12 pioneers and promotes the creation and distribution of high-quality, adaptive online textbooks that can be mixed, modified and printed (i.e., the FlexBook® textbooks). Copyright © 2016 CK-12 Foundation, The names “CK-12” and “CK12” and associated logos and the terms “FlexBook®” and “FlexBook Platform®” (collectively “CK-12 Marks”) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution-Non-Commercial 3.0 Unported (CC BY-NC 3.0) License ( licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated herein by this reference. Complete terms can be found at terms-of-use. Printed: September 13, 2016 AUTHOR SFDR Algebra I Chapter 1. Solving Equations C HAPTER 1 Solving Equations C HAPTER O UTLINE 1.1 Solving Two Step Equations 1.2 Solving Equations by Combining Like Terms 1.3 Solving Equations Using the Distributive Property 1.4 Solving Equations with a Variable on Both Sides 1.5 Solving Rational Equations 1.6 Solving Literal Equations 1.7 Chapter 1 Review 1 1.1. Solving Two Step Equations 1.1 Solving Two Step Equations You will be able to solve a two-step equation for the value of an unknown variable. 8.8.C, A.5.A MEDIA Click image to the left or use the URL below. URL: 2 Chapter 1. Solving Equations Vocabulary Equation = a mathematical statement which shows that two expressions are equal Inverse Operations = opposite operations that undo each other. Addition and subtraction are inverse operations. Multiplication and Division are inverse operations. Independent Practice. TABLE 1.1: 1. 5r + 2 = 17 4. −3 f + 19 = 4 7. 3y - 8 = 1 10. 12.5 = 2g − 3.5 13. 97 = 2n + 19 16. 0.6x + 1.5 = 4 2. 25 = −2w − 3 5. −22 = −x − 12 8. 32 h − 14 = 13 11. 6.3 = 2x − 4.5 14. −9y − 4.2 = 13.8 3. −7 = 4y + 9 6. 3y − 8 = 1 −1 3 9. −2 5 = 4 m+ 5 y 12. −6 = 5 = 4 15. −1 = b4 − 7 PLIX (Play Learn Interact eXplore) T-Shirt Equation Two Step Equations 3 1.2. Solving Equations by Combining Like Terms 1.2 Solving Equations by Combining Like Terms You will be able to solve an equation by combining like terms. 8.8.C, A.5.A FIGURE 1.1 FIGURE 1.2 4 Chapter 1. Solving Equations FIGURE 1.3 MEDIA Click image to the left or use the URL below. URL: Vocabulary Like Terms = terms whose variables and exponents are the same. In other words, terms that are "like" each other. Combine Like Terms = a mathematical process in which like terms are added or subtracted in order to simplify the expression or equation. 5 1.2. Solving Equations by Combining Like Terms Independent Practice Solve each equation. TABLE 1.2: 1. 6a + 5a = −11 3. −3 + 6 − 3x = −18 5. 0 = −5n − 2n 7. 43 = 8.3m + 13.2m 9. 4x + 6 + 3 = 17 11. 5x + 8 − 5x = 8 13. 76 y + 25 + 35 + 71 y = 10 15. 6.25x − 41 x + 2.25 + 7.3x = 0 PLIX (Play Learm Interact eXplore) Multi-Step Equations with Like Terms: Shipments You may also like... Solving Combining Like Terms • • • • • 6 Equation Expression Variable Like terms Inverse operations 2. −6n − 2n = 16 4. x + 11 + 8x = 29 6. −10 = −14v + 14v 8. a − 2 + 3 = −2 10. −10p + 9p = 12 12. 43 x − 1 + 12 x = 11 14. 7q + 4 − 3q − 7 + 5q = 15 16. 0.5y + 5 − 5y + 7 + 4.5y = 0.25 Chapter 1. Solving Equations 1.3 Solving Equations Using the Distributive Property You will solve an equation using the distributive property. FIGURE 1.4 FIGURE 1.5 7 1.3. Solving Equations Using the Distributive Property FIGURE 1.6 MEDIA Click image to the left or use the URL below. URL: Vocabulary Distributive Property = the mathematical law which states that a(b + c) = ab + ac. 8 Chapter 1. Solving Equations Independent Practice Solve each equation. TABLE 1.3: 1. 5(3x + 10) = 215 3. 68 = 4(3b − 1) 5. −185 = 5(3x − 4) 7. 22 = 2(4h − 9) 9. 5(−10 − 6 f ) = −290 11. 4(y − 5) = −15 13. 5(x + 6) − 3 = 37 15. −(n − 8) + 10 = −2 17. 10(1 + 3b) + 15b = −20 19. 23 (6x + 9) − x − 2 = −17 2. 6(5y − 2) = 18 4. −7(x − 3.5) = 0 6. − 13 (x + 6) = 21 8. −6(2w + 4) = −84 10. 42 = 7(6 − 7x) 12. 32 (3x − 12) = 10 14. 5(1 + 4m) − 2m = −13 16. 8 = 8v − 4(v + 8) 18. −5 − 8(1 + 7n) = −8 20. −7.2 + 2(2.5x − 4) = 12 PLIX (Play Learn Interact eXplore) Multi-Step Equations You May Also Like... Multi-Step Equations • • • • • • • Equation Expression Solve Simplify Inverse operations Variable Like Terms 9 1.4. Solving Equations with a Variable on Both Sides 1.4 Solving Equations with a Variable on Both Sides You will use distribution to solve equations with variables on both sides. FIGURE 1.8 FIGURE 1.7 FIGURE 1.8 10 Chapter 1. Solving Equations FIGURE 1.9 MEDIA Click image to the left or use the URL below. URL: 11 1.4. Solving Equations with a Variable on Both Sides Independent Practice. Solve each equation. TABLE 1.4: 1. 5x − 17 = 4x + 36 3. −3y + 8 = 2y − 2 5. −2a + 6 = 30 − 5a 7. 6y − 8 = 1 + 9y 9. 5x + 6 = 5x − 10 11. −3x + 9 = 9 − 3x 13. 6y = −1 + 6y 15. 2(3p + 5) + p = 13 − 2p + 15 17. 2(3b − 4) = 8b − 11 19. 8s − 10 = 27 − (3s − 7) PLIX (Play Learn Interact eXplore) Rubber Ducky Math You May Also Like... Equations with Variables on Both Sides • • • • • 12 Inverse Operations Solve Variable Like terms Distribution Property 2. 36 + 19c = 24c + 6 4. 4 + 6p = −8p + 32 6. 6x − 7 = 4x + 1 8. −14g − 8 = −10g + 40 10. 6p + 2 = −3p − 1 12. 10x = 2x − 16 14. −5m + 2 + 4m = −2m + 11 16. −3y − 10 = 4(y + 2) + 2y 18. −6(2x + 1) = −3x + 7 − 9x 20. 3b + 12 = 3(b − 6) + 4 Chapter 1. Solving Equations 1.5 Solving Rational Equations You will learn how to solve an equation that involves fractions. FIGURE 1.10 FIGURE 1.11 13 1.5. Solving Rational Equations FIGURE 1.12 14 Chapter 1. Solving Equations Vocabulary Rational Equation = an equation in which one or more of the terms is a fractional one. Independent Practice Solve the following rational equations using cross products. TABLE 1.5: 1. 3 c = 4 c−3 2. 1 x−1 =3 3. 2 r = 2 2−r 4. 5 x+3 = 2 x 5. −4 x−1 2 x 6. 3 c+2 = 2 c+2 8. 2 j+4 = 4 j−1 7. 4 = 9. 5 y−3 = 8 x+2 = 11. 2 −x−5 13. −2 x+5 −8 y−4 = = 3 −2x−3 −1 2−x 10. −2 −b+5 12. 6 x+1 = −3 3−x 14. −4 1+x = −3 5−3x = 1 b−2 15 Chapter 1. Solving Equations You May Also Like... Rational Equations Using Proportions • Rational Numbers • Cross Products • Distributive Property 17 1.6. Solving Literal Equations 1.6 Solving Literal Equations You will learn how to solve for any specified variable in any given formula. FIGURE 1.13 FIGURE 1.14 18 Chapter 1. Solving Equations FIGURE 1.15 Vocabulary Literal Equation = an equation made up of mostly letters or variables. Independent Practice Solve the following equations. TABLE 1.6: 1. 5 = x + y 3. a + b = 3 5. p + t = q 7. A = lw 9. d = rt 11. dc = ∏ Solve for x Solve for a Solve for p Solve for w Solve for t Solve for c 2. w = x + 5 4. a + b = 3 6. a2 + b2 = c2 8. A = ∏r2 m 10. r = 2p 12. dc = ∏ Solve for x Solve for b Solve for c2 Solve for ∏ Solve for m Solve for d • Formula • Multi-variable • Literal equations 19 1.7. Chapter 1 Review 1.7 Chapter 1 Review Mixed Review Solve the following equations for the unknown variable. TABLE 1.7: 1. 6x − 1.3 = 3.2 3. 35 x + 25 = 23 5. 1.3x − 0.7x = 12 7. 3(x − 1) − 2(x + 3) = 0 9. A = (b1+b2) h Solve for h 2 11. 42x + 12 = 5x − 3 3 13. x+1 = 2x 15. 2.3x + 2(0.75x − 3.5) = 7.5 2 17. 5(q−7) 12 = 3 19. 0.1(3.2 + 2x) + 0.5(3 − 0.2x) = 0 21. 3(x + 3)− 2(x − 1) = 0 23. 2 a − 13 = 52 a + 23 25. 72 t + 23 = 15 t − 23 20 2. 4(x + 3) = 1 4. 10y + 5 = 10 6. −10a − 2(a + 5) = 14 8. 3x + 6 = x + 15 10. x−3 5 =7 3 2 12. x+1 = x−2 14. C=∏d Solve for d 16. 9(x − 2) − 3x = 3 18. 12x − 16 − 14x − 21 = 3 −4 3 20. x−4 = x+1 22. p = 2l +2w Solve for w 24. 2 5a − 31 = 27 26. P = 4s Solve for s Ch. 2: Relations and Functions SFDR Algebra I Say Thanks to the Authors Click (No sign in required) To access a customizable version of this book, as well as other interactive content, visit CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-source, collaborative, and web-based compilation model, CK-12 pioneers and promotes the creation and distribution of high-quality, adaptive online textbooks that can be mixed, modified and printed (i.e., the FlexBook® textbooks). Copyright © 2016 CK-12 Foundation, The names “CK-12” and “CK12” and associated logos and the terms “FlexBook®” and “FlexBook Platform®” (collectively “CK-12 Marks”) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution-Non-Commercial 3.0 Unported (CC BY-NC 3.0) License ( licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated herein by this reference. Complete terms can be found at terms-of-use. Printed: September 19, 2016 AUTHOR SFDR Algebra I Chapter 1. Relations and Functions C HAPTER 1 Relations and Functions C H AP TE R O U TL I NE 1.1 Identifying Attributes of Relations and Functions 1.2 Domain and Range from Continuous Graphs 1.3 Parent Functions 1.4 Evaluating Functions 1.5 Arithmetic Sequences 1.6 Chapter 2 Review 1 1.1. Identifying Attributes of Relations and Functions 1.1 Identifying Attributes of Relations and Functions You will be able to identify attributes of a function. FIGURE 1.1 2 Chapter 1. Relations and Functions Vocabulary Relation = a collection or set of ordered pairs. Function = a special relationship where each input has a single output and is often written as f(x) where x is the input value. Domain = input values, x values, independent values Range = output values, y values, dependent values Independent Practice Given the relation, identify the domain and range and determine if the relation is a function. 1. {(8, 2), (-4, 1), (-6, 2), (1, 9)} 2. 3. 4. {(1. 3), (1, 0), (1, -2), (1, 8)} 5. 3 1.1. Identifying Attributes of Relations and Functions 6. 7. 8. 9. 10. 11. 4 12. Chapter 1. Relations and Functions {(2, 4), (3, 7), (6, 2), (5, 8), (6, 10)} 13. 14. 15. FIGURE 1.14 • • • • • • Function Function notation Relation Mapping Table Graph 5 1.1. Identifying Attributes of Relations and Functions 1.2 Domain and Range from Continuous Graphs You will be able to identify the reasonable domain and the range of real-world situations and represent them with a graph or an inequality. 6 Chapter 1. Relations and Functions Vocabulary Continuous Graph = a graph which allows x-values to be ANY points including fractions and decimals and is not restricted to defined separate values. Independent Practice Identify if the following graphs are functions or not, also determine their domain and range. TABLE 1.1: 1. 2. 7 1.2. Domain and Range from Continuous Graphs TABLE 1.1: (continued) • • • • • • • • 8 3. 4. 5. 6. 7. 8. 9. 10. Domain Range Continuous Discrete Open circle Closed circle Set notation Real numbers Chapter 1. Relations and Functions 1.3 Parent Functions You will be able to determine the effect on the graphs of the linear and quadratic parent functions when specific values are changed. 9 1.3. Parent Functions Vocabulary Parent Function = the simplest function of a family of functions. Independent Practice Graph the points, then name and describe the parent function of the tables below. TABLE 1.2: 1. 2. 3. 4. 5. 6. 10 Chapter 1. Relations and Functions Name the parent function of the table and set of ordered pairs shown below: 7. 8. {(-1. -5), (0, -2), (1, 1), (2, 4), (3, 7)} Name the parent function of the graphs shown below: TABLE 1.3: 9. 10. Name the parent function of the given mappings shown below: TABLE 1.4: 11. 12. Name the parent function: 13. Hi, I’m the parent of every graph that is a line. 14. Hi, all my kids have a graph that looks like a U. 15. Hi, my graph consist of the points (-1,-1), (0,0) and (1,1). 16. Hi, my graph contains the points (-2,4), (0,0) and (2,4). 17. Hi, I have a domain and range of all real numbers. 18. Hi, I have a domain of all real numbers and a range of all real numbers greater or equal to 0. 11 1.3. Parent Functions • • • • • 12 Parent Function Linear Quadratic Parabola Origin Chapter 1. Relations and Functions 1.4 Evaluating Functions You will use substitution to evaluate an equation. FIGURE 1.24 FIGURE 1.25 13 1.4. Evaluating Functions FIGURE 1.26 Independent Practice 1. If y = 5x + 7, what is the value of y when x = 2? 2. If f(x) = -4x - 12 + x, what is the value of f(x) when x = 3? 3. If y = 6 - 2x, what is the value of x when y = 10? 4. If f(t) = 3(-2t + 7), what is the value of t when f(t) = 39? 5. If f(s) = 2(3s - 4) - 2s + 1, what is the value of f(s) when s = 5? 6. If y = -(5x + 1) + 8x - 4, what is the value of x when y = 22? 7. If (3, 8) is a point on the line whose equation is y = 2x + n, determine the value of n. 8. If (2, 7) is a point on the line whose equation is y = 9. If (4, -10) is a point on the line whose equation is y = h + 7x, determine the value of h. 10. If (-5, 2) is a point on the line whose equation is y = x - m8 , determine the value of m. 11. If (-1, 9) is a point on the line whose equation is f(x) = -2x + p, determine the value of p. 12. If (3, 5) is a point on the line whose equation is f(x) = 5x - c, determine the value of c. • Substitution • Evaluate • Value 14 w 5 +x determine the value of w. Chapter 1. Relations and Functions 1.5 Arithmetic Sequences You will be able to identify and use the pattern of a sequence to find the nth term. FIGURE 1.27 FIGURE 1.28 15 FIGURE 1.29 Vocabulary Arithmetic Sequence = A pattern in which each term is equal to the previous term, plus or minus a constant. The constant is called the common difference (d). 16 Chapter 1. Relations and Functions Independent Practice Determine if the following sequences are arithmetic sequences. Explain. TABLE 1.5: 1. 7, 10, 13, 16 3. 3, 7, 9, 12 2. -19, -15, -11, -7 4. -4, -3, 0, 3, 4 Determine if the following sequences are arithmetic. If so, find the next three terms. TABLE 1.6: 5. -4, -2, 0, 2 7. 8.7, 10.2, 11.7, 13.2 6. 1.5, 2, 2.3, 3.5 8. 13.25, 13.5, 13.75, 14 Find the rule for each one of the following arithmetic sequences. TABLE 1.7: 9. 3, 6, 9, 12 11. -20, -13, -6, 1 10. 39, 32, 25, 18 12. -34, -64, -94, -124 Find the indicated term for each of the following arithmetic sequences. TABLE 1.8: 13. 14. 15. 16. Find the 25th term. Find the 56th term. Find the 13th term. Find the 100th term. -4, -1.5, 1, 3.5 -3, 0, 3, 6 18, 15, 12, 9 23, 28, 33, 38 TABLE 1.9: 17. 18. 19. 20. a1 = 14 a1 = 12 a1 = -6 a1 = 80 d=3 d=-2 d = -8 d = 13 Find the 24th term. Find the 30th term. Find the 11th term. Find the 20th term. • Sequence • Arithmetic sequence • Geometric sequence 17 1.6 Chapter 2 Review Chapter 2 Review Give the domain and range and determine if the relation is a function. 1. {(3, 2), (4, 6), (-5, 7), (-4, 8)} 2. {(1, 12), (1, 14), (1, 16), (1, 18)} 3. {(-2, -12), (0, 6), (1, -3), (4, 6)} Give the relation, domain, and range. Explain if the relation is a function. TABLE 1.10: 4. 5. Determine if the following graphs are functions. Give the domain and range and determine if the graphs are discrete or continuous. TABLE 1.11: 6. 7. 8. 9. 18 1.6. Chapter 2 Review Name the parent function of the following functions TABLE 1.12: 10. 16. 17. 18. 19. TABLE 1.13: Evaluate the following equations. 26 Chapter 1. Relations and Functions TABLE 1.14: Determine if the following sequences are arithmetic by finding the common difference. If so, find the next three terms. Find the indicated term given the first term and the common difference. 27 Linear Functions SFDR Algebra I Andrew Gloag Eve Rawley Anne Gloag Say Thanks to the Authors Click (No sign in required) To access a customizable version of this book, as well as other interactive content, visit CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-source, collaborative, and web-based compilation model, CK-12 pioneers and promotes the creation and distribution of high-quality, adaptive online textbooks that can be mixed, modified and printed (i.e., the FlexBook® textbooks). Copyright © 2016 CK-12 Foundation, The names “CK-12” and “CK12” and associated logos and the terms “FlexBook®” and “FlexBook Platform®” (collectively “CK-12 Marks”) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution-Non-Commercial 3.0 Unported (CC BY-NC 3.0) License ( licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated herein by this reference. Complete terms can be found at terms-of-use. Printed: September 21, 2016 AUTHORS SFDR Algebra I Andrew Gloag Eve Rawley Anne Gloag Chapter 3. Linear Functions C HAPTER 3 Linear Functions C H AP TE R O U TL I NE 1.1 X and Y Intercepts of Linear Functions 1.2 Slope of Linear Functions 1.3 Graphing Linear Functions 1.4 Applications Using Linear Functions 1.5 Writing Linear Functions 1.6 Transformations of Linear Functions 1.7 Scatter Plots 1.8 Direct Variation. 1.9 Chapter 3 Review 1 3.1. X and Y Intercepts of Linear Functions 3.1 X and Y Intercepts of Linear Functions You will learn how to identify and/or solve for the x-intercept and the y-intercept of a function. FIGURE 1.1 FIGURE 1.2 2 Chapter 3. Linear Functions FIGURE 1.3 FIGURE 1.4 3 3.1. X and Y Intercepts of Linear Functions FIGURE 1.5 FIGURE 1.6 4 Chapter 3. Linear Functions Vocabulary x-Intercept = the point at which a line crosses the x axis and the y value is 0. y-Intercept = the point at which a line crosses the y axis and the x value is 0. Independent Practice Find the x and y intercepts. Write your answers as order pairs. TABLE 1.1: TABLE 1.2: 4. 5. 6. 7. Find the x and y intercepts from the following equations. TABLE 1.3: 8. 3x - y = 3 9. 3y - 2x = 6 10. 2x = 4y - 8 5 3.1. X and Y Intercepts of Linear Functions Find the x and y intercepts from the following equations and graph the line. TABLE 1.4: 11. 8x - 3y = 24 12. 5x - 4y = 20 Using intercepts in real world situations. 15. 16. 6 13. 7x + 3y = 21 14. -7x + 2y = 14 Chapter 3. Linear Functions 21. 7 3.1. X and Y Intercepts of Linear Functions 22. • • • • • • 8 intercepts x-intercept y-intercept x-axis y-axis zero Chapter 3. Linear Functions 3.2 Slope of Linear Functions You will be able to identify and calculate the slope of a real world situation given a table, a set of ordered pairs, an equation, a graph or context. FIG...
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