Test Bank for Algebra for College Students, 8th Edition

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INSTRUCTOR’SRESOURCEMANUALALGEBRAFORCOLLEGESTUDENTSEIGHTHEDITIONRobert BlitzerMiami Dade College

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Instructor’s Resource Manual with TestsAlgebra for College Students, Eighth EditionRobert BlitzerTABLE OF CONTENTSMINI-LECTURES(per section)ML-1Chapter 1ML-1Chapter 2ML-11Chapter 3ML-19Chapter 4ML-25Chapter 5ML-32Chapter 6ML-43Chapter 7ML-52Chapter 8ML-61Chapter 9ML-67Chapter 10ML-75Chapter 11ML-83Chapter 12ML-88Mini-Lectures AnswersIncluded at end of sectionADDITIONALEXERCISES(per section)AE-1Chapter 1AE-1Chapter 2AE-46Chapter 3AE-76Chapter 4AE-115Chapter 5AE-151Chapter 6AE-196Chapter 7AE-250Chapter 8AE-295Chapter 9AE-292Chapter 10AE-367Chapter 11AE-399Chapter 12AE-421Additional Exercises AnswersAE-457GROUP ACTIVITIES(per chapter)A-1Chapter 1A-1Chapter 2A-2Chapter 3A-3Chapter 4A-4Chapter 5A-5Chapter 6A-6Chapter 7A-7Chapter 8A-8Chapter 9A-9Chapter 10A-10Chapter 11A-11Chapter 12A-12Group Activities AnswersA-13

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TESTFORMSCHAPTER1TESTS(6TESTS)T-1Form A (FR)T-1Form B (FR)T-3Form C (FR)T-6Form D (MC)T-9Form E (MC)T-11Form F (MC)T-14CHAPTER2TESTS(6TESTS)T-17Form A (FR)T-17Form B (FR)T-21Form C (FR)T-25Form D (MC)T-29Form E (MC)T-34Form F (MC)T-39CUMULATIVEREVIEW1-2(2TESTS)T-44Form A (FR)T-44Form B (MC)T-47CHAPTER3TESTS(6TESTS)T-51Form A (FR)T-51Form B (FR)T-54Form C (FR)T-57Form D (MC)T-60Form E (MC)T-64Form F (MC)T-68CHAPTER4TESTS(6TESTS)T-72Form A (FR)T-72Form B (FR)T-76Form C (FR)T-80Form D (MC)T-84Form E (MC)T-89Form F (MC)T-94CUMULATIVEREVIEW1-4(2TESTS)T-99Form A (FR)T-99Form B (MC)T-102CHAPTER5TESTS(6TESTS)T-107Form A (FR)T-107Form B (FR)T-110Form C (FR)T-113Form D (MC)T-116Form E (MC)T-119Form F (MC)T-122CHAPTER6TESTS(6TESTS)T-125Form A (FR)T-125Form B (FR)T-127Form C (FR)T-129Form D (MC)T-131Form E (MC)T-134Form F (MC)T-137CUMULATIVEREVIEW1-6(2TESTS)T-140Form A (FR)T-140Form B (MC)T-143

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CHAPTER7TESTS(6TESTS)T-148Form A (FR)T-148Form B (FR)T-151Form C (FR)T-154Form D (MC)T-157Form E (MC)T-161Form F (MC)T-164CHAPTER8TESTS(6TESTS)T-167Form A (FR)T-167Form B (FR)T-171Form C (FR)T-175Form D (MC)T-178Form E (MC)T-182Form F (MC)T-186CUMULATIVEREVIEW1-8(2TESTS)T-190Form A (FR)T-190Form B (MC)T-193CHAPTER9TESTS(6TESTS)T-197Form A (FR)T-197Form B (FR)T-200Form C (FR)T-203Form D (MC)T-206Form E (MC)T-210Form F (MC)T-214CHAPTER10TESTS(6TESTS)T-218Form A (FR)T-218Form B (FR)T-222Form C (FR)T-226Form D (MC)T-230Form E (MC)T-235Form F (MC)T-239CUMULATIVEREVIEW1-10(2TESTS)T-242Form A (FR)T-242Form B (MC)T-245CHAPTER11TESTS(6TESTS)T-249Form A (FR)T-249Form B (FR)T-251Form C (FR)T-253Form D (MC)T-255Form E (MC)T-257Form F (MC)T-259CHAPTER12TESTS(6TESTS)T-273Form A (FR)T-273Form B (FR)T-275Form C (FR)T-277Form D (MC)T-279Form E (MC)T-281Form F (MC)T-283FINAL(2TESTS)T-286Form A (FR)T-286Form B (MC)T-292

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TESTANSWERKEYST-299Chapter 1T-299Chapter 2T-301Cumulative Review 1-2T-305Chapter 3T-306Chapter 4T-309Cumulative Review 1-4T-313Chapter 5T-315Chapter 6T-317Cumulative Review 1-6T-318Chapter 7T-319Chapter 8T-321Cumulative Review 1-8T-324Chapter 9T-325Chapter 10T-327Cumulative Review 1-10T-331Chapter 11T-332Chapter 12T-336FinalsT-338

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ML-1Mini Lecture 1.1Algebraic Expressions, Real Numbers and Interval NotationLearning Objectives:1.Translate English phrases into algebraic expressions.2.Evaluate algebraic expressions.3.Use mathematical models.4.Recognize the sets that make up the real numbers.5.Use set-builder notation.6.Use the symbols∈and∉.7.Use inequality symbols.8.Use interval notation.Examples:1.Write each English phrase as an algebraic expression. Letxrepresent the number.a.Three less than five times a number.b.The product of a number and six, increased by four.2.Evaluate each algebraic expression for the given value or values of the variable(s).a.352++xx, forx= 2b.)(22yxx++, forx= 3y= 43.Use the roster method to list the elements in each set.a.{x|xis an integer between 4 and 9}b.{x|xis an even whole number less than 10}4.Use the meaning of the symbols∈and∉to determine whether each statement is true orfalse.a.3∈{x|xis a natural number}b.9∉{1, 3, 5, 7}5.Write out the meaning of each inequality. Then determine whether the inequality is trueor false.a. –10 > –8b.02–≤c.33≥d.5–2≤6.Express the interval[)5,-¥in set builder notation and graph.Teaching Notes:•Be sure to go over important vocabulary for the section including: variable, algebraicexpression, constant, exponential expression, equation, formula, natural numbers, wholenumbers, integers, rational, irrational and real numbers.•Brainstorm the many words that translate to the four basic operations. Ex: increased -addition.•nb= b· b· … b (b appears as a factor “n” times).•Order of operation rules include:1.First, perform all operations within grouping symbols.2.Evaluate all exponential expressions.3.Do all multiplication and division in the order in which they occur, working from leftto right.4.Last, do all additions and subtractions in the order in which they occur, working fromleft to right.•< is read “less than”, > is read “greater than”•≤ is read “less than or equal to”, ≥ is read “greater than or equal to”

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ML-2Answers: 1. a.3–5xb.46+x2. a. 17b. 233. a. {5, 6, 7, 8}b. {0, 2, 4, 6, 8}4. a. trueb. true 5. a.–10 greater than –8,falseb.–2 is less than or equal to 0,truec. 3 is greater than or equal to 3,trued. 2 is less than or equal to –5,false6.{}|5xx³ -

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ML-3Mini Lecture 1.2Operations With Real Numbers and Simplifying Algebraic ExpressionsLearning Objectives:1.Find a number’s absolute value.2.Add real numbers.3.Find opposites.4.Subtract real numbers.5.Multiply real numbers.6.Evaluate exponential expressions.7.Divide real numbers.8.Use the order of operations.9.Use commutative, associative, and distributive properties.10.Simplify algebraic expressions.Examples:1.Find the absolute value.a.8–b.43–c.24.6–d.122.Add or subtract.a. –14 + 25b.31–43–+c. 15 – (–10)d. 6.8 – 12.32e.83–85–f. –52 + 52g. –32 – (–38)h. 4.2 – (–8.1)3.Evaluate.a.()28–b.28–c.()43–d.43–4.Multiply or divide.a.20853–÷b. (15) (–1) (–4)c.024–d. (–3.3) (1.2)e.( )( )( )()2076−f.180g.328−÷−h.151473−⋅5.Use the distributive property and simplify.a. 6(x–2)b. –3 (6 –y)c. –4(x– 5 –y)6.Rewrite to show how the associative property could be used to simplify the expression.Then simplify.a. 6(–4x)b. (x+ 124) + 3767.Simplify using the order of operation.a.64205+÷⋅b.22–15)2(5–)2(–6c.()6 323xx--

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ML-4Teaching Notes:•Remind students that absolute value measures distance from zero, and for that reason, itis always positive.•Opposites and additives inverses are just different names for the same thing.•Students need to be reminded often that a negative sign is only part of the base if it isinside parentheses with the base.•When opposites are added, the result is zero.•Make sure students understand what is behind subtraction – why subtraction can bechanged to addition of the opposite.•Never, never, never multiply the base and the exponent together! Students are oftentempted to do this.Answers: 1. a. 8b.43c. 6.24d. 122. a. 11b.1211–1213–orc. 25d. –5.52e. –1f. 0g. 6h. 12.33. a. 64b. –64c. 81d. –814. a.23–b. 60c. undefinedd. –3.96 e. 0 f. 0 g. 12h.52−5. a. 6x– 12b. –18 + 3yc. –4x+ 20 + 4y6. a.xx24–)4–6(=⋅b.500)376124(+=++xx7. a. 31 b. –2c. 15x– 12

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ML-5Mini Lecture 1.3Graphing EquationsLearning Objectives:1.Plot points in the rectangular coordinate system.2.Graph equations in the rectangular coordinate system.3.Use the rectangular system to visualize relationships between variables.4.Interpret information about a graphing utility’s viewing rectangle or table.Examples:1.Plot the following point in a rectangular coordinate system.A. (–2, 3)B. (–4, 0)C. (1, 5)D. (–1, –4)E. (3, –3)F. (0, 2)2.Complete the table of values for3–xy=, then graph the equation.3.Complete the table of values for2–2xy=, then graph the equation.4.Complete the table of values for,1–xy=then graph the equation.xy= |x–1|(x,y)–4–3–2–101234xy=x– 3(x,y)–2–1012xy= 2 –x2(x,y)–3–2–10123

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ML-6Teaching Notes:•The point plotting method is one method for graphing equations.•The graph of a linear equation is a line.•The graph of a quadratic equation is a parabola.•The graph of an absolute value equation is a “V” shape that can shoot upward ordownward.Answers: 1.2. (–2, –5)(–1, –4)(0, –3)(1, –2)(2, –1)3. (–3, –7)(–2, –2)(–1, 1)(0, 2)(1, 1)(2, –2)(3, –7)4. (–4, 5)(–3, 4)(–2, 3)(–1, 2)(0, 1)(1, 0)(2, 1)(3, 2)(4, 3)Figure for Answer 3Figure for Answer 4

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ML-7Mini Lecture 1.4Solving Linear EquationsLearningObjectives:1.Solve linear equations.2.Recognize identities, conditional equations, and inconsistent equations.3.Solve applied problems using mathematical models.Examples:Solve each equation. If fractions are involved, you may want to clear the fractions first.1.a.2275=+xb.xx7332=+c.9–23=+x2.a.)1–2(4)23(5xx=+b.)4–(5)6–2(–)4–(5xxx=c.)5–(3–25xx=3.a.107–54xx=b.653–2–81=+aac.151–3–34–=yyd.)5–3(51)82(31xx=+Solve and determine whether the equation is an identity, a conditional equation, or aninconsistent equation.4.a.xxx3)4–(235+=+b.)4(488)4–2(6++=+xxxc.3)1(5)4(23++=++aaad.)33(8)44(6+=+yye.14–4.110–6.0xx=f.0)–2(3)1–(6=+xxTeaching Notes:•Students may need to be reminded there is no “right” or “wrong” side of the equation.Some students have a problem when the variable ends up on the right side of theequation.•Students need to practice clearing equations of fractions by multiplying each term(whether it is a fraction or not) by the least common denominator of all the terms.Answers:1. a. 3b. 8c. –62. a. –2b. 3c. 103. a. 10b. 3c. –8d. –554. a. Inconsistent ; NoSolutionb. Inconsistent; No Solutionc. Identity; infinitely many solutionsd. Identity; infinitely manysolutionse. 5; conditional equationf. 0; conditional equation

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ML-8Mini Lecture 1.5Problem Solving and Using FormulasLearning Objectives:1.Solve algebraic word problems using linear equations.2.Solve a formula for a variable.Examples:Solve the following using the five step strategy for solving word problems.1.When 12 is subtracted from three times a number, the result is 36. What is thenumber?2.15% of what number is 255?3.In a triangle, the measure of the third angle is twice the measure of the first angle.The measure of the second angle is twenty more than the first. Find the measure ofeach angle.4.The dog run is six feet longer than it is wide and the perimeter measures 32 feet.Determine the measurements of the length and width of the dog run.5.A new automobile sells for $28,000. If the mark-up is 25% of the dealer’s cost, whatis the dealer’s cost?Solve each formula for the specified variable.6.BhV31=forB7.)(rtlPA+=forP8.)2–(180nS=forn9.2dKMmf=forM10.CByAx=+forATeaching Notes:•Use the five step strategy for solving word problems.•Read the problem carefully. Let a variable represent one of the quantities in the problem.•If necessary, write an expression for any other unknown quantities in the problem interms of the same variable used in step 1.•Write an equation to describe the conditions of the problem.•Solve the equation and answer the problem’s question.•Remind students to always check to make sure their answer makes sense.Answers: 1.16;3612–3==xx2.1700;25515.0==xx3.80,60,40;1802)20(=+++xxx4.longfeet116x,feet wide5;32)6(22=+==++xxx5.400,22$;000,2825.0==+xxx6.hVB3=7.rtlAP+=8.2180+=Sn9.KmfdM2=10.xByCA–=

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ML-9Mini Lecture 1.6Properties of Integral ExponentsLearning Objectives:1.Use the product rule.2.Use the quotient rule.3.Use the zero-exponent rule.4.Use the negative exponent rule.5.Use the power rule.6.Find the power of a product.7.Find the power of a quotient.8.Simplify exponential expressions.Examples:Simplify. Final answers should not contain any negative exponents.1. a.34yy⋅b.)3)(4(4aac.)4)(–(34xyzzxyd.)6)((3–35321nmnm2. a.410yyb.26525aac.6–3xxd.2–4264618zyxzyx3. a.2–4b.3–51c.06d.06xe.3–2–3yxf.5–47yxg.8–2–aah.1–154. a.210)(xb.3–6–)(yc.1–2)4(d.24)(a5. a.202)3(bab.2–2–3–)5(yxc.2–32d.23–226yx6. a.223–6)4)(5(yxyxb.2–44–2–2–3)6()2(yxyxc.2–3–34)2(5yxyxd.3–641baTeaching Notes:•Exponent rules are very easy as presented – one at a time. Students often becomeconfused when several rules are used in one problem. Constant reinforcement and lots ofpractice will help.•Remind students that when a variable appears to have no exponent – there is an invisibleexponent of one.•Never, never, never multiply a base and an exponent together.•Always (exception: scientific notation) write final answers with positive exponents only.Answers: 1. a.7yb.512ac.4524–zyxd.263nm2. a.6yb.45ac.9xd.3223zyx3. a.161b. 125c. 1d. 6e.323yxf.547yxg.6ah.1514. a.20xb.18yc.161d.8a5. a.49ab.2546yxc.49d.649yx6. a.yx1080b.14129xyc.2520xyd.1812ba

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ML-10Mini Lecture 1.7Scientific NotationLearning Objectives:1.Convert from scientific to decimal notation.2.Convert from decimal to scientific notation.3.Perform computations with scientific notation.4.Use scientific notation to solve problems.Examples:1.Write each number in decimal notation.a.5104.3–×b.4–10015.2×2.Write each number in scientific notation.a. 32,500,000,000b. –0.00417c.4109432×3.Perform the indicated computations, writing the answers in scientific notation.a.)108)(104.2(5–3××b.2–4104108.6××4.In Central City, the population is 176,000. Express the population in scientific notation.Teaching Notes:•A number is written in scientific notation when it is expressed in the formna10×with1 ≤ |a| < 10 and “n” is an integer.•When multiplying terms written in scientific notationmnmnbaba+××=××10)()10)(10(.•When dividing terms written in scientific notation.101010–nmnmbaba×=××•When multiplying or dividing is complete, make sure the final answer is in scientificnotation.•Students need to be reminded that a number must be written as a number between 1 and10 to be in scientific notation.•The sign of a number has nothing to do with the sign of the power when a number iswritten in scientific notation.Answers:1. a. –340,000b. 0.00020152. a.101025.3×b.3–1017.4×c.710432.9×3. a.1–1092.1×b.6107.1×4.51076.1×

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ML-11Mini Lecture 2.1Introduction to FunctionsLearning Objectives:1.Find the domain and range of a relation.2.Determine whether a relation is a function.3.Evaluate a function.Examples:1.Find the domain and range of the relation.a. {(1, 5), (2, 10), (3, 15), (4, 20), (5, 25)}b. {(1, -1), (0, 0), (-5, 5)}2.Determine whether each relation is a function.a.{})10,5(),9,5(),8,5(),7,5(),6,5(b. {(5, 6), (6, 7), (7, 8), (8, 9), (9, 10)}3.Find the indicated function value.a.2–3)(for)3(xxff=b.4–2)(for)2(–2+=xxxggc.23–)(for)1(–2+=ttthhd.32)(for)(+=+xxfhaf4.Functiongis defined by the tableFind the indicated function value.a.g(2)b.g(4)Teaching Notes:•Arelationis any set of ordered pairs.•The set of all first terms “x-values” of the ordered pairs is called thedomain.•The set of all second terms “y-values” of the ordered pairs is called therange.•Afunctionis a relation in which each member of the domain corresponds to exactly onemember of the range.•A function is a relation in which no two ordered pairs have the same first component anddifferent second components.•The variable “x” is called theindependent variablebecause it can be assigned any valuefrom the domain.•The variable “y” is called thedependent variablebecause its value depends on “x”.•The notationf(x), read “fofx” represents the value of the function at the number “x”.Answers: 1. domain {1, 0, -5} range {-1, 0, 5}2. a. not a functionb. function3. a. 7b. 14c. 6d.322++han4. a. 6 b. 10xg (x)02142638410

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ML-12XY-6-4-2246-6-4-22460Mini Lecture 2.2Graphs of FunctionsLearning Objectives:1.Graph functions by plotting points.2.Use the vertical line test to identify functions.3.Obtain information about a function from its graph.4.Identify the domain and range of a function from its graph.Examples:State the domain of each function.1.Graph the function13)(and3)(+==xxgxxfin the same rectangular coordinatesystem. Graph integers forxstarting with –2 and ending with 2. How is the graph ofgrelated to the graph off?2.Use the vertical line test to identify graphs in whichyis a function ofx.a.b.c.3.Use the graph offto find the indicated function value.a.f(2)b.f(0)c.f(1)4.Use the graph each function to identify its domain and range.a.b.XY-6-4-2246-6-4-22460(-5 ,1)(1, 1)(-2, 1)(4, 1)XY-6-4-2246-6-4-22460

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ML-13Teaching notes:•The graph of a function is the graph of the ordered pairs.•If a vertical line intersects a graph in more than one point, the graph does not defineyas a function ofx.Answers:1. The graph ofgis the graph offshifted up 1 unit.2. a. yesb. noc. yes 3. a. 0 b. 4 c. 14. a. Domain:{}5,1, 1.4-Range:{ }1b. Domain:[)0,¥Range:[)2,¥

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ML-14Mini Lecture 2.3The Algebra of FunctionsLearning Objectives:1.Find the domain of a function.2.Use the algebra of functions to combine functions and determine domains.Examples:State the domain of each function.1.a.( )31fxx=-b.( )42xg xx=-c.( )26h xxx=+-d.( )1759p xxx=++-2.Let( )xxxf22−=and( )3+=xxg. Find the following;a.()( )xgf+b. the domain ofgf+c.()()2−+gf3.Let( )25+=xxfand( )61g xx=-. Find the following;a.()( )xgf+b. The domain ofgf+4.Let( )12+=xxfand( )3=+xxg. Find the following;a.()( )xgf+b.()()2−+gfc.()( )xgf−d.()( )0gf−e.()2−gfTeaching Notes:•Students need to be reminded that division by zero is undefined. The value of “x” cannotbe anything that would make the denominator of a fraction zero.•Students often exclude values from the domain that would make the numerator zero,warn against this.•Show students why the radicand of a square root function must be greater than or equal tozero. This is a good place to use the graphing calculator so students can “see” whathappens.Answers: 1. a.(),-¥ ¥b.()(), 2or2,-¥¥c.()(), 6or6,-¥¥d.()()(),5or5, 9or9,-¥ --¥2. a.23xx-+b.(),-¥ ¥c. 34. a.5621xx++-b.()()(),2or2, 1or1,-¥ --¥4. a.22−+xxb. 0c.42+−xxd. 4 e. -1

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ML-15Mini Lecture 2.4Linear Functions and SlopeLearning Objectives:1.Use intercepts to graph a linear function in standard form.2.Compute a line’s slope.3.Find a line’s slope andy-intercept from its equation.4.Graph linear functions in slope-intercept form.5.Graph horizontal or vertical lines.6.Interpret slope as rate of change.7.Find a function’s average rate of change.8.Use slope andy-intercept to model data.Examples:1.Use intercepts and a checkpoint to graph each linear function. Name thex-intercept and they-intercept.a.1052=+yxb.– 24xy=2.Find the slope of the line passing through each pair of points. Then indicate whether the linethrough the points rises, falls, is horizontal, or is vertical.a.(2, 5) and (–6, 3)b.)3,1(and)0,5(c.(3, 0) and (3,4)-d.(2, 4) and (6, 4)-3.a. Find the slope andy-intercept for the line whose equation is1243=+yxand then graphthe equation.b. Find the slope andy-intercept for the linear function321)(+=xxfand then graph thefunction.4.Graph the linear equations.a.2=xb.12–3=yTeaching Notes:•The standard form of the equation of a line isCByAx=+, as long asAandBare notboth zero.•Ax-intercept will have a correspondingycoordinate of 0.•Ay-intercept will have a correspondingxcoordinate of 0.•The slope of a line compares the vertical change to the horizontal change()runrise.•Slope formula is:1212––xxyym=.•A line that rises from left to right has a positive slope.•A line that falls from left to right has a negative slope.•A line that is horizontal has zero slope.•A line that is vertical has an undefined slope.•The slope-intercept form of the equation of a line isbmxy+=wheremis the slope andbis they-intercept.

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ML-16Answers: 1. a.b.2. a.41=m, risesb.34m= -, fallsc. undefined, verticald. 0, horizontal3. a.3)(0,intercept-43–343–ymxy=+=b.12m= -y-intercept (0,3)4. a.b.

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ML-17Mini Lecture 2.5The Point-Slope Form of the Equation of a LineLearning Objectives:1.Use the point-slope form to write equations of a line.2.Model data with linear functions and make predictions.3.Find slopes and equations of parallel and perpendicular lines.Examples:1.Write an equation in point-slope form for the line with the given information.a. slope = –3, passing through (–4, 1)b. slope = 32 , passing through (6, 3)c. slope = 0, passing through (–3, 2)2.Write an equation in point slope form for the line with the given information. Then writethe equation in slope-intercept form.a. slope = 41 , passing through (4, 3)b. passing through (1, 5) and (3, –5)c. passing through (–2, 4) and (2, 6)3.Find the slope of a line parallel to each given line.a.743+=xyb.63–2=yxc.82=+yx4.Find the slope of a line perpendicular to each given line.a.14–+=xyb.123=+yxc.102–4=yxWrite an equation for each of the following in slope-intercept form.5.A line passing through the origin (0, 0) and parallel to a line whose equation is5–4xy=.6.A line passing through (4, –4) and parallel to a line whose equation is62=+yx.7.A line passing through (6, –1) and perpendicular to a line whose equation is13–=yx.8.A line passing through (2, –5) and perpendicular to a line whose equation is072–4=+xy.Teaching Notes:•Thepoint-slope form of the equationof a non-vertical line with slopemthat passesthrough the point()yx,is()11xxmyy−=−.•Make sure students memorize the point-slope form of a linear equation and know whateach letter represents.•Students will need lots of practice on this.•This is a good time for students to be able to visualize parallel line lines and “see” thatthey have the same slopes, but differenty-intercepts.

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ML-18Answers:1. a.)4(3–1–+=xyb.6–(323–xy=)c.)3(02–+=xy2.a.)4–(413–xy=;241+=xyb.)1–(5–5–xy=or)3–(5–5xy=+;105–+=xyc.)2–(216–xy=or)2(214–+=xy;521+=xy3. a.43b.32c.21–4. a.41b.31c.21–5.xy4=6.2–21–xy=7.173–+=xy8.1–2–xy=

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ML-19Mini Lecture 3.1Systems of Linear Equations in Two VariablesLearning Objectives:1.Determine whether an ordered pair is a solution of a system of linear equations.2.Solve systems of linear equations by graphing.3.Solve systems of linear equations by substitution.4.Solve systems of linear equations by addition.5.Select the most efficient method for solving a system of linear equations.6.Identify systems that do not have exactly one ordered-pair solution.Examples:1.Determine whether (2, –3) is a solution of the system.2421xyxyì+=ïïíï+= -ïî2.Solve by graphing:–302xyyxì+=ïïíï=+ïî3.Solve by the substitution method.a.29349yxxyì=+ïïíï-=ïîb.3512411xyxyì+=ïïíï+=ïî4.Solve by the addition method.a.2–9354xyxyì+=ïïíï+=ïîb.53421266xyxy+=== -5.Solve by the method of your choice. Identify inconsistent systems and systems withdependent equations.a.– 21362xyxyì=ïïíï-=ïîb.3––3626xyxyì=ïïíï-==ïîTeaching Notes:•A system of linear equations in two variables represents a pair of lines. There are threepossibilities for solutions:a. If two lines intersect at one point, then there is exactly one ordered-pair solution.b. If two lines are parallel, then there is no solution.c. If two lines are identical, then there are infinitely many solutions.•All three methods for solving systems of linear equations in two variables will producethe same answer; however, one method will sometimes be more efficient than another.Answers: 1. no2. (1, 3)3. a. (–9, –9)b. (–1, 3)4. a. (–7, 5)b. (–1, 2)5. a. Inconsistent; the lines are parallel, no solutionb. {(x,y) | 3x–y= –3} or {(x,y) | –6x+ 2y= 6}the lines coincide and the system has infinitely many solutions.

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Test Bank for Algebra for College Students, 8th Edition - Page 26 preview image

ML-20Mini Lecture 3.2Problem Solving and Business Applications Using Systems of EquationsLearning Objectives:1.Solve problems using systems of equations.2.Use functions to model revenue, cost, and profit, and perform a break even analysis.Examples:1.At JoJo’s pet and supply store female gerbils sell for $8 each and male gerbils sell for $5each. On Saturday 20 gerbils were sold for a total of $139. How many males and how manyfemales were sold on Saturday?2.To promote the grand opening of Leroy’s Toys the store gave away 9000 miniature cars andgoofy sunglasses. The cars cost 13¢ and the sunglasses cost 15¢ each. Leroy spent a total of$1290 on the giveaways. How many of each item did he buy?3.The larger of two numbers is equal to three times the smaller. If twice the larger is added tothree times the smaller, the sum is 27. Find the numbers.4.Twice the length of a rectangle is equal to five times its width. The perimeter of the rectangleis 77 inches. Find the dimensions of the rectangle.5.Cashews cost $3.60 per pound and almonds cost $2.70 per pound. For a fundraiser, thevolleyball team will be selling bags of mixed nuts. How many pounds of cashews and howmany pounds of almonds should the team buy in order to make a 60 pound mixture that willsell for $3.00 per pound?6.How many gallons of 15% alcohol solution and how many gallons of 40% alcohol solutionshould be mixed to get 20 gallons of a 30% alcohol solution?7.Joe and Jack inherited $200,000 from their Aunt Lulu. They each decided to put their moneyinto savings accounts for 1 year and then decide how to spend it. Joe’s money earned 5%interest and Jack’s earned 3.8%. Together, their money earned $8,608 in interest. How muchdid each boy inherit?8.A small airplane can travel 600 miles in 4 hours with the wind. The return trip against thewind takes 5 hours. Find the speed of the plan in still air and the speed of the wind.9.Since Jane’s grandparents enjoy making birdhouses and selling them at local craft shows.They will pay $150 for the booth rental for the weekend. The materials for making eachbirdhouse cost $8.75. If they are able to sell the birdhouses for $15.00 each, how many willthey need to sell to beak even? What if they sell 15 birdhouses? 40 birdhouses?Answers: 1. 13 females; 7 males2. 6000 sunglasses; 3000 cars3. 3 and 94. 11 inches wide;27.5 inches long5. 20 lbs. cashews’ 40 lbs. almonds6. 12 gallons of 40%; 8 gallons of 15%7. Joe $84,000; Jack $116,0008. 135 mph; 15 mph wind 9. 24 birdhouses to break even; they willlose $56.25; they will make $100.

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Test Bank for Algebra for College Students, 8th Edition - Page 27 preview image

ML-21Mini Lecture 3.3Systems of Linear Equations in Three VariablesLearning Objectives:1.Verify the solution of a system of linear equations in three variables.2.Solve systems of linear equations in three variables.3.Identify inconsistent and dependent systems.4.Solve problems using systems in three variables.Examples1.Show that the ordered triple (1, 2, 3) is a solution of the system:623234xyzxyzxyzì++=ïïïï-+=íïï+-= -ïïî2.Solve the system:a.42222xyzxyzxyzì++=ïïïï--=íïï+-=ïïîb.–4– 32286130xyzxyzxyzì+=ïïïï-+=íïï-+=ïïîc.23–546210435xyzxyzxyzì+=ïïïï+-=íïï-+=ïïî3.Create three equations from the stated problem and then solve.The sum of the three numbers is 14. The largest is 4 times the smallest, while the sum ofthe smallest and twice the largest is 18.4.Find the quadratic function2yaxbxc===whose graph passes through the points(1, 3), (2, 5), and (–1, 11).Teaching Notes:•A system of linear equations is three variables represents three planes.•A linear system that intersects at one point is called aconsistent systemand has anordered triple as an answer (x, y, z).•A linear system that intersects at infinitely many points is also called aconsistent systemand is also calleddependent.•A linear system that has no common point(s) of intersection represents aninconsistentsystemand has no solution.Answers: 1. 1+2+3=6 , 2–2+3=3 , 1+4–9= –4 2. a. (3,–1,2)b. No Solution , Inconsistent system.c. infinitely many solutions, dependant equations 3.14=++zyx,xz4=,182=+zx. Thenumbers are 2, 4 and 8.4.2245yxx=-+or( )2245fxxx=-+

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Test Bank for Algebra for College Students, 8th Edition - Page 28 preview image

ML-22Mini Lecture 3.4Matrix Solutions to Linear SystemsLearning Objectives:1.Write the augmented matrix for a linear system.2.Perform matrix row operations.3.Use matrices to solve linear systems in two variables.4.Use matrices to solve linear systems in three variables.5.Use matrices to identify inconsistent and dependent systems.Examples:1.Write the matrix for each system.a.3– 41265xyxyì=ïïíï+=ïîb.2748xyxì+=ïïíï=ïîc.2303432510xyzxyzxyzì++=ïïïï-+=íïï+-=ïïî2.Write the system of linear equations represented by each augmented matrix.a.111121–1b.06113–2c.1–11–32212–11–1143.Perform each matrix row operation as indicated and write the new matrix.a.4–13213–2b.134–3–221c.8–252–3112121–2–21RR↔212–RR+21212andRRRR+↔4.Solve each system using matrices.a.310121xyxyì+=ïïíï+= -ïîb.35335229xyxxyzxyzì++=ïïïï+-=íïï+-=ïïîc.22123240xyzxyzxyzì++=ïïïï-++=íïï++=ïïîTeaching Notes:•Organization and neatness is very important when using matrices to solve systems.•Caution students to watch signs carefully.•Using the calculator is a great way to check systems solved using matrices.•Students tend to panic if fractions “happen”. Encourage them to keep working throughthe problem.

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ML-23Answers: 1. a.512614–3b.870412c.10305–1214–33212. a.1121–=+=yxyxb.063–2=+=yxyxc.1–2211–2––34=++=+=++zyxzyxzyx3. a.134–3–221b.214–7–021c.8–922–314301214. a. (–3, 1)b. (1, 2, –2)c. (–2, 3, –1)

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ML-24Mini Lecture 3.5Determinants and Cramer’s RuleLearning Objectives:1.Evaluate a second-order determinant.2.Solve a system of linear equations in two variables using Cramer’s rule.3.Evaluate a third-order determinant.4.Solve a system of linear equations in three variables using Cramer’s rule.5.Use determinants to identify inconsistent and dependent systems.Examples:1.Evaluate the determinant of each of the following matrices:a.3201b.52–3–82.Use Cramer’s rule to solve the system:2– 334210xyxyì=ïïíï-=ïî3.Evaluate the determinant of the following matrix:2102–0101–24.Use Cramer’s rule to solve the system:–3627234xyzxyzxyzì++=ïïïï++=íïï++=ïïî5.Use Cramer’s rule to solve each system or to determine that the system is inconsistent orcontains dependent equations.a.2– 4523xyxyì=ïïíï-+=ïîb.261638xyxyì+=ïïíï+=ïîTeaching Notes:•A matrix of ordernm×hasmrows andncolumns.•The determinant of a 2 x 2 matrix2211babais denoted by2211baba=1221–baba.•If the determinant D1= 0 and at least one of the determinants in the numerator is not 0,then the system is inconsistent and there is no solution .•If the determinant, D1= 0 and all the determinants in the numerators are 0, then theequations in the systems are dependent and the system has infinitely many solutions.Answers:1. a. 3b. 342. (3, 1)3. 64. (2, –1, 3)5. a. inconsistentb. dependent equations;infinitely many solutions

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Test Bank for Algebra for College Students, 8th Edition - Page 31 preview image

ML-25Mini Lecture 4.1Solving Linear InequalitiesLearning Objectives:1.Solve linear inequalities.2.Recognize inequalities with no solution or all real numbers as solutions.3.Solve applied problems using linear inequalities.Examples:1.Solve and graph the solution set on a number line. Give solutions in interval notation.a.2x–5 ≥3b.3x– 5 ≤ 6x+ 4c.41+x>41–2x+ 83d.4(x+ 1) > 4x+ 2e.2x+ 2 ≤ 2x– 2Teaching Notes:•One method of representing the solution set of an inequality is withinterval notation.Using the notationx≥ –3 is expressed as [–3, ∞). The bracket indicates –3 is included inthe interval. The infinity symbol does not represent a real number and the intervalextends indefinitely to the right.•When multiplying or dividing both sides of an inequality by a negative quantity,remember to reverse the direction of the inequality symbol.•When an inequality has been solved and the variable has been eliminated and the result isa false statement, the inequality has no solution, Ø.•When an inequality has been solved and the variable has been eliminated and the result isa true statement, the solution for the inequality is all real numbers.Answers:1. a. [4, ∞)b. [–3, ∞)c. (–∞,21)d. (–∞, ∞)e. Ø

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