Systems of Equations and Matrices

Transcription

1 Sstems of Equations and Matrices A sstem of equations is a collection of two or more variables In this chapter, ou should learn the following How to use the methods of substitution and elimination to solve sstems of linear equations in two variables () How to solve multivariable linear sstems () How to solve sstems of inequalities () How to use matrices to solve sstems of linear equations () How to perform operations with matrices () How to find inverses of matrices and use inverse matrices to solve sstems of linear equations () How to find determinants of square matrices (7) Graham Hewood / istockphotocom How can ou use a matri to model the number of people in the United States who participate in snowboarding? (See Section, Eercise ) The graphs above show the three possible tpes of solutions for a sstem of two linear equations in two variables: infinitel man solutions, no solution, and one solution (See Section ) 8

2 8 Chapter Sstems of Equations and Matrices Sstems of Linear Equations in Two Variables Use the method of substitution to solve sstems of equations in two variables Use the method of elimination to solve sstems of linear equations in two variables Interpret graphicall the numbers of solutions of sstems of linear equations in two variables Use sstems of linear equations in two variables to model and solve real-life problems The Method of Substitution Up to this point in the tet, most problems have involved either a function of one variable or a single equation in two variables However, man problems in science, business, and engineering involve two or more equations in two or more variables To solve such problems, ou need to find solutions of a sstem of equations Here is an eample of a sstem of two equations in two unknowns Equation Equation A solution of this sstem is an ordered pair that satisfies each equation in the sstem Finding the set of all solutions is called solving the sstem of equations For instance, the ordered pair, is a solution of this sstem To check this, ou can substitute for and for in each equation Check (, ) in Equation and Equation :?? Write Equation Substitute for and for Solution checks in Equation Write Equation Substitute for and for Solution checks in Equation In this chapter, ou will stud four was to solve sstems of equations, beginning with the method of substitution The guidelines for solving a sstem of equations b the method of substitution are summarized below GUIDELINES FOR SOLVING A SYSTEM OF EQUATIONS BY THE METHOD OF SUBSTITUTION Solve one of the equations for one variable in terms of the other Substitute the epression found in Step into the other equation to obtain an equation in one variable Solve the equation obtained in Step Back-substitute the value obtained in Step into the epression obtained in Step to find the value of the other variable Check that the solution satisfies each of the original equations

3 Sstems of Linear Equations in Two Variables 8 The term back-substitution implies that ou work backwards First ou solve for one of the variables, and then ou substitute that value back into one of the equations in the sstem to find the value of the other variable The back-substitution reduces the two-equation sstem to one equation in a single variable EXAMPLE Solving a Sstem of Equations b Substitution EXPLORATION Use a graphing utilit to graph and in the same viewing window Use the zoom and trace features to find the coordinates of the point of intersection What is the relationship between the point of intersection and the solution found in Eample? Solve the sstem of equations Equation Equation Solution Begin b solving for in Equation Solve for in Equation Net, substitute this epression for into Equation and solve the resulting single-variable equation for Write Equation Substitute for Distributive Propert Combine like terms Divide each side b Finall, ou can solve for b back-substituting into the equation, to obtain Write revised Equation Substitute for Solve for The solution is the ordered pair, You can check this solution as follows Check Substitute, into Equation : Write Equation? Substitute for and Solution checks in Equation Substitute, into Equation :? Write Equation Substitute for and Solution checks in Equation Because, satisfies both equations in the sstem, it is a solution of the sstem of equations STUDY TIP Because man steps are required to solve a sstem of equations, it is ver eas to make errors in arithmetic So, ou should alwas check our solution b substituting it into each equation in the original sstem

4 8 Chapter Sstems of Equations and Matrices The equations in Eample are linear The method of substitution can also be used to solve sstems in which one or both of the equations are nonlinear Such a sstem ma have more than one solution EXPLORATION Use a graphing utilit to graph the two equations in Eample 7 in the same viewing window How man solutions do ou think this sstem has? Repeat this eperiment for the equations in Eample How man solutions does this sstem have? Eplain our reasoning EXAMPLE Substitution: Two-Solution Case Solve the sstem of equations 7 Solution Begin b solving for in Equation to obtain Net, substitute this epression for into Equation and solve for Equation Equation Substitute for in Equation Simplif Write in general form Factor, Solve for Back-substituting these values of to solve for the corresponding values of produces the solutions, and, Check these in the original sstem The sstem of equations in Eample has two solutions It is possible that a sstem has no solutions, as shown in Eample EXAMPLE Substitution: No Real-Solution Case Solve the sstem of equations Solution Begin b solving for in Equation to obtain Net, substitute this epression for into Equation and solve for ± ± Equation Equation Substitute for in Equation Simplif Quadratic Formula Simplif Because the discriminant is negative, the equation has no (real) solution So, the original sstem has no (real) solution

5 Sstems of Linear Equations in Two Variables 8 The Method of Elimination So far, ou have studied one method for solving a sstem of equations: substitution Now ou will stud the method of elimination The ke step in this method is to obtain, for one of the variables, coefficients that differ onl in sign so that adding the equations eliminates the variable 7 Equation Equation Add equations Note that b adding the two equations, ou eliminate the -terms and obtain a single equation in Solving this equation for produces, which ou can then back-substitute into one of the original equations to solve for EXAMPLE Solving a Sstem of Equations b Elimination Solve the sstem of linear equations Equation Equation Solution Because the coefficients of differ onl in sign, ou can eliminate the -terms b adding the two equations Write Equation Write Equation 8 Add equations Solve for B back-substituting into Equation, ou can solve for Write Equation Substitute for Simplif Solve for The solution is, Check this in the original sstem, as follows Check?? Substitute into Equation Equation checks Substitute into Equation Equation checks NOTE Although ou could use either the method of substitution or the method of elimination to solve the sstem in Eample, ou ma find that the method of elimination is more efficient

6 8 Chapter Sstems of Equations and Matrices EXAMPLE Solving a Sstem of Equations b Elimination STUDY TIP To obtain coefficients (for one of the variables) that differ onl in sign, ou often need to multipl one or both of the equations b suitabl chosen constants Solve the sstem of linear equations 7 Equation Equation Solution For this sstem, ou can obtain coefficients of the -terms that differ onl in sign b multipling Equation b 7 7 Write Equation Multipl Equation b Add equations So, ou can see that B back-substituting this value of into Equation, ou can solve for 7 7 Write Equation Substitute for Combine like terms Solve for The solution is, Check this in both equations in the original sstem Check? 7 7? Substitute into Equation Equation checks Substitute into Equation Equation checks In Eample, the two sstems of linear equations 7 and 7 are called equivalent sstems because the have precisel the same solution set The operations that can be performed on a sstem of linear equations to produce an equivalent sstem are () interchanging an two equations, () multipling an equation b a nonzero constant, and () adding a multiple of one equation to an other equation in the sstem GUIDELINES FOR SOLVING A SYSTEM OF EQUATIONS BY THE METHOD OF ELIMINATION Obtain coefficients for or that differ onl in sign b multipling all terms of one or both equations b suitabl chosen constants Add the equations to eliminate one variable and solve the resulting equation Back-substitute the value obtained in Step into either of the original equations and solve for the other variable Check that the solution satisfies each of the original equations

7 Sstems of Linear Equations in Two Variables 87 Graphical Interpretation of Solutions It is possible for a general sstem of equations to have eactl one solution, two or more solutions, or no solution If a sstem of linear equations has two different solutions, it must have an infinite number of solutions GRAPHICAL INTERPRETATIONS OF SOLUTIONS For a sstem of two linear equations in two variables, the number of solutions is one of the following Number of Solutions Graphical Interpretation Slopes of Lines Eactl one solution The two lines intersect at one point The slopes of the two lines are not equal Infinitel man solutions The two lines coincide (are identical) The slopes of the two lines are equal No solution The two lines are parallel The slopes of the two lines are equal A sstem of linear equations is consistent if it has at least one solution A consistent sstem with eactl one solution is independent, whereas a consistent sstem with infinitel man solutions is dependent A sstem is inconsistent if it has no solution EXAMPLE Recognizing Graphs of Linear Sstems Match each sstem of linear equations with its graph Describe the number of solutions and state whether the sstem is consistent or inconsistent a b c i ii iii Solution a The graph of sstem (a) is a pair of parallel lines (ii) The lines have no point of intersection, so the sstem has no solution The sstem is inconsistent b The graph of sstem (b) is a pair of intersecting lines (iii) The lines have one point of intersection, so the sstem has eactl one solution The sstem is consistent c The graph of sstem (c) is a pair of lines that coincide (i) The lines have infinitel man points of intersection, so the sstem has infinitel man solutions The sstem is consistent STUDY TIP A comparison of the slopes of two lines gives useful information about the number of solutions of the corresponding sstem of equations To solve a sstem of equations graphicall, it helps to begin b writing the equations in slope-intercept form Tr doing this for the sstems in Eample

8 88 Chapter Sstems of Equations and Matrices In Eamples 7 and 8, note how ou can use the method of elimination to determine that a sstem of linear equations has no solution or infinitel man solutions EXAMPLE 7 No-Solution Case: Method of Elimination + = = Figure Solve the sstem of linear equations Equation Equation Solution To obtain coefficients that differ onl in sign, ou can multipl Equation b 7 Multipl Equation b Write Equation False statement Because there are no values of and for which 7, ou can conclude that the sstem is inconsistent and has no solution The lines corresponding to the two equations in this sstem are shown in Figure Note that the two lines are parallel and therefore have no point of intersection In Eample 7, note that the occurrence of a false statement, such as 7, indicates that the sstem has no solution In the net eample, note that the occurrence of a statement that is true for all values of the variables, such as, indicates that the sstem has infinitel man solutions EXAMPLE 8 Man-Solution Case: Method of Elimination Solve the sstem of linear equations (, ) (, ) = Equation Equation Solution To obtain coefficients that differ onl in sign, ou can multipl Equation b Multipl Equation b Write Equation Add equations Because the two equations are equivalent (have the same solution set), ou can conclude that the sstem has infinitel man solutions The solution set consists of all points, ling on the line Figure as shown in Figure Letting a, where a is an real number, ou can see that the solutions of the sstem are a, a In Eample 8, choose some values of a to find solutions of the sstem: for eample, if a, the solution is,, and if a, the solution is, Then check these solutions in the original sstem

9 Sstems of Linear Equations in Two Variables 89 Applications At this point, ou ma be asking the question How can I tell which application problems can be solved using a sstem of linear equations? The answer comes from the following considerations Does the problem involve more than one unknown quantit? Are there two (or more) equations or conditions to be satisfied? If one or both of these situations occur, the appropriate mathematical model for the problem ma be a sstem of linear equations Original flight WIND r r Return flight r + r Figure WIND EXAMPLE 9 An Application of a Linear Sstem An airplane fling into a headwind travels the -mile fling distance between Chicopee, Massachusetts and Salt Lake Cit, Utah in hours and minutes On the return flight, the same distance is traveled in hours Find the airspeed of the plane and the speed of the wind, assuming that both remain constant Solution The two unknown quantities are the speeds of the wind and the plane If r is the speed of the plane and r is the speed of the wind, then r r speed of the plane against the wind r r speed of the plane with the wind as shown in Figure Using the formula distance ratetime for these two speeds, ou obtain the following equations r r r r These two equations simplif as follows r r r r To solve this sstem b elimination, multipl Equation b Equation Equation r r r r r r r r Write Equation Multipl Equation b So, and r, 777 miles per hour r 7 miles per hour, r Check this solution in the original statement of the problem Add equations Speed of plane Speed of wind

10 87 Chapter Sstems of Equations and Matrices In a free market, the demands for man products are related to the prices of the products As the prices decrease, the demands b consumers increase and the amounts that producers are able or willing to suppl decrease Price per unit (in dollars) p (,,, ) Demand Suppl 7,,,, Number of units Figure EXAMPLE Finding the Equilibrium Point The demand and suppl equations for a new tpe of personal digital assistant are p Demand equation p Suppl equation where p is the price in dollars and represents the number of units Find the equilibrium point for this market The equilibrium point is the price p and number of units that satisf both the demand and suppl equations Solution Because p is written in terms of, begin b substituting the value of p given in the suppl equation into the demand equation p 9,, Write demand equation Substitute for p Combine like terms Solve for So, the equilibrium point occurs when the demand and suppl are each million units (See Figure ) The price that corresponds to this -value is obtained b back-substituting,, into either of the original equations For instance, back-substituting into the demand equation produces p,, $ The solution is,,, Check this in both equations in the original sstem Eercises See wwwcalcchatcom for worked-out solutions to odd-numbered eercises In Eercises 8, fill in the blanks A set of two or more equations in two or more variables is called a of A of a sstem of equations is an ordered pair that satisfies each equation in the sstem Finding the set of all solutions to a sstem of equations is called the sstem of equations The first step in solving a sstem of equations b the method of is to solve one of the equations for one variable in terms of the other variable The first step in solving a sstem of equations b the method of is to obtain coefficients for (or ) that differ onl in sign Two sstems of equations that have the same solution set are called sstems 7 A sstem of linear equations that has at least one solution is called, whereas a sstem of linear equations that has no solution is called 8 In business applications, the is defined as the price p and the number of units that satisf both the demand and suppl equations In Eercises 9, determine whether each ordered pair is a solution of the sstem of equations 9 (a), (b), (c) (d),, (a), (b), 9 (c) (d) 7,, 7 e (a), (b), 7 (c), (d), (a) 9, 7 9 (b), log (c), (d),

11 Sstems of Linear Equations in Two Variables 87 In Eercises 8, solve the sstem b the method of substitution In Eercises 9, solve the sstem b the method of elimination Label each line with its equation To print an enlarged cop of the graph, go to the website wwwmathgraphscom In Eercises 7, solve the sstem b the method of elimination and check an solutions analticall u v u v b m b m 8 r s r s

12 87 Chapter Sstems of Equations and Matrices In Eercises 7 7, use a graphing utilit to graph the lines in the sstem Use the graphs to determine if the sstem is consistent or inconsistent If the sstem is consistent, determine the number of solutions Then solve the sstem if possible In Eercises 7 78, use an method to solve the sstem WRITING ABOUT CONCEPTS 79 What is meant b a solution of a sstem of equations in two variables? 8 When solving a sstem of equations b substitution, how do ou recognize that the sstem has no solution? 8 Write a brief paragraph describing an advantages of substitution over the graphical method of solving a sstem of equations 8 Find equations of lines whose graphs intersect the graph of the parabola at (a) two points, (b) one point, and (c) no points (There is more than one correct answer) Use graphs to support our answer WRITING ABOUT CONCEPTS (continued) In Eercises 8 and 8, the graphs of the two equations appear to be parallel Yet, when the sstem is solved analticall, ou find that the sstem does have a solution Find the solution and eplain wh it does not appear on the portion of the graph that is shown Briefl eplain whether or not it is possible for a consistent sstem of linear equations to have eactl two solutions 8 Give eamples of a sstem of linear equations that has (a) no solution and (b) an infinite number of solutions 87 Consider the sstem of equations a b c d e f (a) Find values for a, b, c, d, e, and f so that the sstem has one distinct solution (There is more than one correct answer) (b) Eplain how to solve the sstem in part (a) b the method of substitution and graphicall (c) Write a brief paragraph describing an advantages of the method of substitution over the graphical method of solving a sstem of equations CAPSTONE 88 Rewrite each sstem of equations in slope-intercept form and sketch the graph of each sstem What is the relationship among the slopes of the two lines, the number of points of intersection, and the number of solutions? (a) (b) 8 (c) 8

13 Sstems of Linear Equations in Two Variables 87 Suppl and Demand In Eercises 89 9, find the equilibrium point of the demand and suppl equations The equilibrium point is the price p and number of units that satisf both the demand and suppl equations Demand 89 p 9 p 9 p 9 p 9 DVD Rentals The weekl rentals for a newl released DVD of an animated film at a local video store decreased each week At the same time, the weekl rentals for a newl released DVD of a horror film increased each week Models that approimate the weekl rentals R for each DVD are R R 8 Suppl p 8 p p 8 p Animated film Horror film where represents the number of weeks each DVD was in the store, with corresponding to the first week (a) After how man weeks will the rentals for the two movies be equal? (b) Use a table to solve the sstem of equations numericall Compare our result with that of part (a) 9 Suppl and Demand The suppl and demand curves for a business dealing with wheat are Suppl: p Demand: p 88 7 where p is the price in dollars per bushel and is the quantit in bushels per da Use a graphing utilit to graph the suppl and demand equations and find the market equilibrium (The market equilibrium is the point of intersection of the graphs for > ) 9 Choice of Two Jobs You are offered two jobs selling dental supplies One compan offers a straight commission of % of sales The other compan offers a salar of $ per week plus % of sales How much would ou have to sell in a week in order to make the straight commission offer better? 9 Choice of Two Jobs You are offered two different jobs selling college tetbooks One compan offers an annual salar of $, plus a ear-end bonus of % of our total sales The other compan offers an annual salar of $, plus a ear-end bonus of % of our total sales Determine the annual sales required to make the second offer better 97 Investment Portfolio A total of $, is invested in two funds paing % and 8% simple interest (The % investment has a lower risk) The investor wants a earl interest income of $ from the two investments (a) Write a sstem of equations in which one equation represents the total amount invested and the other equation represents the $ required in interest Let and represent the amounts invested at % and 8%, respectivel (b) Use a graphing utilit to graph the two equations in the same viewing window As the amount invested at % increases, how does the amount invested at 8% change? How does the amount of interest income change? Eplain (c) What amount should be invested at % to meet the requirement of $ per ear in interest? 98 Investment Portfolio A total of $, is invested in two corporate bonds that pa % and % simple interest The investor wants an annual interest income of $9 from the investments What amount should be invested in the % bond? 99 Acid Miture Thirt liters of a % acid solution is obtained b miing a % solution with a % solution (a) Write a sstem of equations in which one equation represents the amount of final miture required and the other represents the percent of acid in the final miture Let and represent the amounts of the % and % solutions, respectivel (b) Use a graphing utilit to graph the two equations in part (a) in the same viewing window As the amount of the % solution increases, how does the amount of the % solution change? (c) How much of each solution is required to obtain the specified concentration of the final miture? Fuel Miture Five hundred gallons of 89-octane gasoline is obtained b miing 87-octane gasoline with 9-octane gasoline (a) Write a sstem of equations in which one equation represents the amount of final miture required and the other represents the amounts of 87- and 9-octane gasolines in the final miture Let and represent the numbers of gallons of 87-octane and 9-octane gasolines, respectivel (b) Use a graphing utilit to graph the two equations in part (a) in the same viewing window As the amount of 87-octane gasoline increases, how does the amount of 9-octane gasoline change? (c) How much of each tpe of gasoline is required to obtain the gallons of 89-octane gasoline?

14 87 Chapter Sstems of Equations and Matrices Geometr What are the dimensions of a rectangular tract of land if its perimeter is kilometers and its area is square kilometers? Geometr What are the dimensions of an isosceles right triangle with a two-inch hpotenuse and an area of square inch? Airplane Speed An airplane fling into a headwind travels the 8-mile fling distance between Pittsburgh, Pennslvania and Phoeni, Arizona in hours and minutes On the return flight, the distance is traveled in hours Find the airspeed of the plane and the speed of the wind, assuming that both remain constant Airplane Speed Two planes start from Los Angeles International Airport and fl in opposite directions The second plane starts hour after the first plane, but its speed is 8 kilometers per hour faster Find the airspeed of each plane if hours after the first plane departs the planes are kilometers apart Nutrition Two cheeseburgers and one small order of French fries from a fast-food restaurant contain a total of 8 calories Three cheeseburgers and two small orders of French fries contain a total of calories Find the caloric content of each item Nutrition One eight-ounce glass of apple juice and one eight-ounce glass of orange juice contain a total of 77 milligrams of vitamin C Two eight-ounce glasses of apple juice and three eight-ounce glasses of orange juice contain a total of 7 milligrams of vitamin C How much vitamin C is in an eight-ounce glass of each tpe of juice? 7 Prescriptions The numbers of prescriptions P (in thousands) filled at two pharmacies from through are shown in the table Year Pharmac A 9 9 Pharmac B 8 (a) Use a graphing utilit to create a scatter plot of the data for pharmac A and use the regression feature to find a linear model Let t represent the ear, with t corresponding to Repeat the procedure for pharmac B (b) Assuming the numbers for the given five ears are representative of future ears, will the number of prescriptions filled at pharmac A ever eceed the number of prescriptions filled at pharmac B? If so, when? 8 Data Analsis A store manager wants to know the demand for a product as a function of the price The dail sales for different prices of the product are shown in the table Price, $ $ $ Demand, 7 (a) Find the least squares regression line for the data b solving the sstem for b 7a 7b 9a 9 a b a and b (b) Use the regression feature of a graphing utilit to confirm the result in part (a) (c) Use the graphing utilit to plot the data and graph the linear model from part (a) in the same viewing window (d) Use the linear model from part (a) to predict the demand when the price is $7 In Eercises 9, find a sstem of linear equations that has the given solution (There is more than one correct answer) 9, 8,,, In Eercises and, find the value of k such that the sstem of linear equations is inconsistent 8 k k 9 True or False? In Eercises, determine whether the statement is true or false If it is false, eplain wh or give an eample that shows it is false In order to solve a sstem of equations b substitution, ou must alwas solve for in one of the two equations and then back-substitute If a sstem consists of a parabola and a circle, then the sstem can have at most two solutions 7 If two lines do not have eactl one point of intersection, then the must be parallel 8 Solving a sstem of equations graphicall will alwas give an eact solution 9 If a sstem of linear equations has no solution, then the lines must be parallel To solve a sstem using the method of elimination, the equations in the sstem must be linear

15 Multivariable Linear Sstems 87 Multivariable Linear Sstems Use back-substitution to solve linear sstems in row-echelon form Use Gaussian elimination to solve sstems of linear equations Solve nonsquare sstems of linear equations Use sstems of linear equations in three or more variables to model and solve real-life problems Row-Echelon Form and Back-Substitution The method of elimination can be applied to a sstem of linear equations in more than two variables In fact, this method easil adapts to computer use for solving linear sstems with dozens of variables When elimination is used to solve a sstem of linear equations, the goal is to rewrite the sstem in a form to which back-substitution can be applied To see how this works, consider the following two sstems of linear equations Sstem of Three Linear Equations in Three Variables: (See Eample ) z 9 z 7 Equivalent Sstem in Row-Echelon Form: (See Eample ) z 9 z z The second sstem is said to be in row-echelon form, which means that it has a stair-step pattern with leading coefficients of After comparing the two sstems, it should be clear that it is easier to solve the sstem in row-echelon form, using back-substitution EXAMPLE Using Back-Substitution in Row-Echelon Form Solve the sstem of linear equations z 9 Equation z Equation z Equation Solution From Equation, ou know the value of z To solve for, substitute z into Equation to obtain Substitute for z Solve for Then substitute and z into Equation to obtain 9 Substitute for and for z Solve for The solution is,, and z, which can be written as the ordered triple,, Check this in all three equations in the original sstem of equations

16 87 Chapter Sstems of Equations and Matrices Christopher Lui/China Stock CHUI-CHANG SUAN-SHU One of the most influential Chinese mathematics books was the Chui-chang suan-shu or Nine Chapters on the Mathematical Art (written in approimatel BC) Chapter Eight of the Nine Chapters contained solutions of sstems of linear equations using positive and negative numbers One such sstem was as follows z 9 z This sstem was solved using column operations on a matri Matrices (plural for matri) will be discussed later in this chapter Gaussian Elimination Two sstems of equations are equivalent if the have the same solution set To solve a sstem that is not in row-echelon form, first convert it to an equivalent sstem that is in row-echelon form b using the following operations OPERATIONS THAT PRODUCE EQUIVALENT SYSTEMS Each of the following row operations on a sstem of linear equations produces an equivalent sstem of linear equations Interchange two equations Multipl one of the equations b a nonzero constant Add a multiple of one of the equations to another equation to replace the latter equation To see how this is done, take another look at the method of elimination, as applied to a sstem of two linear equations EXAMPLE Using Gaussian Elimination to Solve a Sstem Solve the sstem of linear equations Equation Equation Solution There are two strategies that seem reasonable: eliminate the variable or eliminate the variable The following steps show how to use the first strateg Interchange the two equations in the sstem Multipl the first equation b Add the multiple of the first equation to the second equation to obtain a new second equation New sstem in row-echelon form Notice in the first step that interchanging rows is an eas wa of obtaining a leading coefficient of Now back-substitute into Equation and solve for Substitute for Solve for The solution is and which can be written as the ordered pair,

17 Multivariable Linear Sstems 877 Rewriting a sstem of linear equations in row-echelon form usuall involves a chain of equivalent sstems, each of which is obtained b using one of the three basic row operations listed on the previous page This process is called Gaussian elimination, after the German mathematician Carl Friedrich Gauss (777 8) EXAMPLE Using Gaussian Elimination to Solve a Sstem STUDY TIP Arithmetic errors are often made when performing elementar row operations You should note the operation performed in each step so that ou can go back and check our work Solve the sstem of linear equations Equation Equation z 9 z 7 Equation Solution Because the leading coefficient of the first equation is, ou can begin b saving the at the upper left and eliminating the other -terms from the first column z 9 z z 9 z z 7 z 8 z 7 z z 9 z z Write Equation Write Equation Add Equation to Equation Adding the first equation to the second equation produces a new second equation Multipl Equation b Write Equation Add revised Equation to Equation Adding times the first equation to the third equation produces a new third equation Now that all but the first have been eliminated from the first column, go to work on the second column (You need to eliminate from the third equation) z 9 Adding the second equation to the z third equation produces a new z third equation Finall, ou need a coefficient of for z in the third equation z 9 Multipling the third equation b z produces a new third equation z This is the same sstem that was solved in Eample, and, as in that eample, ou can conclude that the solution is,, and z

18 878 Chapter Sstems of Equations and Matrices The net eample involves an inconsistent sstem one that has no solution The ke to recognizing an inconsistent sstem is that at some stage in the elimination process ou obtain a false statement such as EXAMPLE An Inconsistent Sstem (a) Solution: one point (b) Solution: one line (c) Solution: one plane Solve the sstem of linear equations Equation Equation z z z Equation Solution z z z z z z z z Adding times the first equation to the second equation produces a new second equation Adding times the first equation to the third equation produces a new third equation Adding times the second equation to the third equation produces a new third equation Because is a false statement, ou can conclude that this sstem is inconsistent and has no solution Moreover, because this sstem is equivalent to the original sstem, ou can conclude that the original sstem also has no solution As with a sstem of linear equations in two variables, the solution(s) of a sstem of linear equations in more than two variables must fall into one of three categories (d) Solution: none THE NUMBER OF SOLUTIONS OF A LINEAR SYSTEM For a sstem of linear equations, eactl one of the following is true There is eactl one solution There are infinitel man solutions There is no solution (e) Solution: none Figure In Section, ou learned that a sstem of two linear equations in two variables can be represented graphicall as a pair of lines that are intersecting, coincident, or parallel A sstem of three linear equations in three variables has a similar graphical representation it can be represented as three planes in space that intersect in one point (eactl one solution) [see Figure (a)], intersect in a line or a plane (infinitel man solutions) [see Figures (b) and (c)], or have no points common to all three planes (no solution) [see Figures (d) and (e)]

19 Multivariable Linear Sstems 879 EXAMPLE A Sstem with Infinitel Man Solutions Solve the sstem of linear equations Equation Equation z z Equation STUDY TIP In Eample, and are solved in terms of the third variable z To write the correct form of the solution to the sstem that does not use an of the three variables of the sstem, let a represent an real number and let Then solve for and The solution can then be written in terms of a, which is not one of the variables of the sstem z a Solution z z z z z Adding the first equation to the third equation produces a new third equation Adding times the second equation to the third equation produces a new third equation This result means that Equation depends on Equations and in the sense that it gives no additional information about the variables Because is a true statement, ou can conclude that this sstem will have infinitel man solutions However, it is incorrect to sa simpl that the solution is infinite You must also specif the correct form of the solution So, the original sstem is equivalent to the sstem z z In the last equation, solve for in terms of z to obtain z Back-substituting z in the first equation produces z Finall, letting z a, where a is a real number, the solutions to the given sstem are all of the form a, a, and z a So, ever ordered triple of the form a, a, a a is a real number is a solution of the sstem STUDY TIP When comparing descriptions of an infinite solution set, keep in mind that there is more than one wa to describe the set In Eample, there are other was to write the same infinite set of solutions For instance, b solving for and z in terms of and letting b (where b is a real number), the solutions could have been written as b, b, b b is a real number To convince ourself that this description produces the same set of solutions, consider the following Substitution Solution a b a b a b,,,,,,,,,,,,,,,,,,,,,,,, Same Solution Same Solution Same Solution

20 88 Chapter Sstems of Equations and Matrices Nonsquare Sstems So far, each sstem of linear equations ou have looked at has been square, which means that the number of equations is equal to the number of variables In a nonsquare sstem, the number of equations differs from the number of variables A sstem of linear equations cannot have a unique solution unless there are at least as man equations as there are variables in the sstem EXAMPLE A Sstem with Fewer Equations than Variables Solve the sstem of linear equations Equation z z Equation Solution Begin b rewriting the sstem in row-echelon form z z Adding times the first equation to the second equation produces a new second equation z z Multipling the second equation b produces a new second equation Solve for in terms of z to obtain z, and back-substitute z into Equation to obtain z Finall, b letting z a, where a is a real number, ou have the solution a, a, and z a So, ever ordered triple of the form a, a, a, where a is a real number, is a solution of the sstem Because there were originall three variables and onl two equations, the sstem cannot have a unique solution Applications (, ) Figure = (, ) (, ) EXAMPLE 7 Data Analsis: Curve-Fitting Find the equation of the parabola a b c whose graph passes through the points,,,, and, Solution Because the graph of a b c passes through the points,,,, and,, ou can substitute for and in the equation a b c for each ordered pair to produce the following sstem of linear equations a b c Equation : Substitute for and for a b c Equation : Substitute for and for a b c Equation : Substitute for and for The solution of this sstem is a, b, and c So, the equation of the parabola is, as shown in Figure

21 Multivariable Linear Sstems 88 s t = t = t = t = Figure 7 Recall that the height at time t of an object that is moving in a (vertical) line with constant acceleration g is given b the position function st gt v t s where st is the height of the object at time t, v is the initial velocit (at t ), and s is the initial height of the object Eample 8 demonstrates how a sstem of equations can be used to find the position function given the heights at various times EXAMPLE 8 Vertical Motion An object moving verticall is at the following heights at the specified times At t second, s feet At t seconds, s feet At t seconds, s feet Find the values of g, v, and s in the position function st gt v t s (See Figure 7) Solution B substituting t and s into the position function, ou can obtain three linear equations in g, v, and s When t : When t : When t : g v s g v s g v s g v s g v s 9g v s Solving this sstem ields g, v 8, and s, which can be written as, 8, This solution results in a position function of st t 8t and implies that the object was thrown upward at a velocit of 8 feet per second from a height of feet EXAMPLE 9 Investment Analsis An inheritance of $, was invested among three funds: a mone-market fund that paid % annuall, municipal bonds that paid % annuall, and mutual funds that paid 7% annuall The amount invested in mutual funds was $ more than the amount invested in municipal bonds The total interest earned during the first ear was $7 How much was invested in each tpe of fund? Solution Let,, and z represent the amounts invested in the mone-market fund, municipal bonds, and mutual funds, respectivel From the given information, ou can write the following equations z, Equation z Equation 7z 7 Equation Rewriting this sstem in standard form without decimals produces the following Equation Equation z, z, 7z 7, Equation Using Gaussian elimination to solve this sstem ields,, and z 7 So, $ was invested in the mone-market fund, $ was invested in municipal bonds, and $7 was invested in mutual funds

22 88 Chapter Sstems of Equations and Matrices Eercises See wwwcalcchatcom for worked-out solutions to odd-numbered eercises In Eercises, fill in the blanks A sstem of equations that is in form has a stair-step pattern with leading coefficients of A solution to a sstem of three linear equations in three unknowns can be written as an, which has the form,, z The process used to write a sstem of linear equations in row-echelon form is called elimination Interchanging two equations of a sstem of linear equations is a that produces an equivalent sstem A sstem of equations is called if the number of equations differs from the number of variables in the sstem The function st) gt v t s is called the function, and it models the height st of an object at time t that is moving in a vertical line with a constant acceleration g In Eercises 7, determine whether each ordered triple is a solution of the sstem of equations 7 z z 9 z (a),, (b),, (c),, (d),, 8 z 7 z 7z (a),, (b),, (c),, (d),, 9 z 8 z 7 (a) (b) (c),, (d) 8z z 7 (a) (c),, 7,, 8,, 9 (b) (d),,,,,,,, In Eercises, use back-substitution to solve the sstem of linear equations z z 8 z z z z 8 z z z z 8 z z In Eercises 7, solve the sstem of linear equations and check an solution analticall 7 8 z z z 7 z 9 z z 9 z z z 9 z z 7 z 9 z z z 7 z 7 7z z 9 z z 8z 9 z 8z z z z z z z z z 7 z z 7 z z 8z 8 8z 8 z z z 8 9 z z z z 8 z 7 z z 8 z z z 8

23 Multivariable Linear Sstems 88 z z z z 9 z 8 z z z z w w w w z z z z w w w z 7 8 z z z z 7z z 9z z z WRITING ABOUT CONCEPTS In Eercises and, perform the row operation and write the equivalent sstem Add Equation to Equation Equation Equation z z z Equation What did this operation accomplish? Add times Equation to Equation Equation Equation z z z Equation What did this operation accomplish? Are the following two sstems of equations equivalent? Give reasons for our answer z z z z 7 z 7 z One of the following sstems is inconsistent and the other has one solution How can ou identif each b observation? 8 9 WRITING ABOUT CONCEPTS (continued) 7 When using Gaussian elimination to solve a sstem of linear equations, how can ou recognize that the sstem has no solution? Give an eample that illustrates our answer 8 Eplain the graphical significance of a sstem of three equations with three unknowns having a unique solution Vertical Motion In Eercises 9, an object moving verticall is at the given heights at the specified times Find the position function st gt v t s for the object 9 At t second, s 8 feet At t seconds, s 8 feet At t seconds, s feet At t second, s feet At t seconds, s feet At t seconds, s feet At t second, s feet At t seconds, s 7 feet At t seconds, s feet At t second, s feet At t seconds, s feet At t seconds, s feet In Eercises 8, find the equation of the parabola a b c that passes through the points To verif our result, use a graphing utilit to plot the points and graph the parabola,,,,,,,,,,,,,,,,,,,, 7,,,,, 8,,,,, In Eercises 9, find the equation of the circle D E F that passes through the points To verif our result, use a graphing utilit to plot the points and graph the circle 9,,,,,,,,,,,,,,, 8,,,,,

24 88 Chapter Sstems of Equations and Matrices Sports In Super Bowl I, on Januar, 97, the Green Ba Packers defeated the Kansas Cit Chiefs b a score of to The total points scored came from different scoring plas, which were a combination of touchdowns, etra-point kicks, and field goals, worth,, and points, respectivel The same number of touchdowns and etra-point kicks were scored There were si times as man touchdowns as field goals How man touchdowns, etra-point kicks, and field goals were scored during the game? (Source: Super Bowlcom) Sports In the 8 Women s NCAA Final Four Championship game, the Universit of Tennessee Lad Volunteers defeated the Universit of Stanford Cardinal b a score of to 8 The Lad Volunteers won b scoring a combination of two-point baskets, three-point baskets, and one-point free throws The number of two-point baskets was two more than the number of free throws The number of free throws was two more than five times the number of three-point baskets What combination of scoring accounted for the Lad Volunteers points? (Source: National Collegiate Athletic Association) Agriculture A miture of liters of chemical A, liters of chemical B, and liters of chemical C is required to kill a destructive crop insect Commercial spra X contains,, and parts, respectivel, of these chemicals Commercial spra Y contains onl chemical C Commercial spra Z contains onl chemicals A and B in equal amounts How much of each tpe of commercial spra is needed to get the desired miture? Acid Miture A chemist needs liters of a % acid solution The solution is to be mied from three solutions whose concentrations are %, %, and % How man liters of each solution will satisf each condition? (a) Use liters of the % solution (b) Use as little as possible of the % solution (c) Use as much as possible of the % solution 7 Finance A small corporation borrowed $77, to epand its clothing line Some of the mone was borrowed at 8%, some at 9%, and some at % How much was borrowed at each rate if the annual interest owed was $7, and the amount borrowed at 8% was four times the amount borrowed at %? 8 Finance A small corporation borrowed $8, to epand its line of tos Some of the mone was borrowed at 8%, some at 9%, and some at % How much was borrowed at each rate if the annual interest owed was $7, and the amount borrowed at 8% was five times the amount borrowed at %? Investment Portfolio In Eercises 9 and 7, consider an investor with a portfolio totaling $, that is invested in certificates of deposit, municipal bonds, blue-chip stocks, and growth or speculative stocks How much is invested in each tpe of investment? 9 The certificates of deposit pa % annuall, and the municipal bonds pa % annuall Over a five-ear period, the investor epects the blue-chip stocks to return 8% annuall and the growth stocks to return % annuall The investor wants a combined annual return of % and also wants to have onl one-fourth of the portfolio invested in stocks 7 The certificates of deposit pa % annuall, and the municipal bonds pa % annuall Over a five-ear period, the investor epects the blue-chip stocks to return % annuall and the growth stocks to return % annuall The investor wants a combined annual return of % and also wants to have onl one-fourth of the portfolio invested in stocks 7 Pulle Sstem A sstem of pulles is loaded with 8-pound and -pound weights (see figure) The tensions t and t in the ropes and the acceleration a of the -pound weight are found b solving the sstem of equations t t t where t and t are measured in pounds and a is measured in feet per second squared t lb t 8 lb (a) Solve this sstem (b) The -pound weight in the pulle sstem is replaced b a -pound weight The new pulle sstem will be modeled b the following sstem of equations t t t a a t t 8 a a 8 Solve this sstem and use our answer for the acceleration to describe what (if anthing) is happening in the pulle sstem

25 Multivariable Linear Sstems 88 7 Electrical Network Appling Kirchhoff s Laws to the electrical network in the figure, the currents I, I, and I are the solution of the sstem I I I I I 7 I I 8 Find the currents Ω I I Ω 7 Data Analsis: Stopping Distance In testing a new automobile braking sstem, the speed (in miles per hour) and the stopping distance (in feet) were recorded in the table Speed, Stopping Distance, 88 (a) Find a quadratic equation that models the data (b) Graph the model and the data on the same set of aes (c) Use the model to estimate the stopping distance when the speed is 7 miles per hour 7 Data Analsis: Wildlife A wildlife management team studied the reproduction rates of deer in three tracts of a wildlife preserve Each tract contained acres In each tract, the number of females, and the percent of females that had offspring the following ear, were recorded The results are shown in the table Number, Percent, 7 8 I (a) Find a quadratic equation that models the data (b) Use a graphing utilit to graph the model and the data in the same viewing window (c) Use the model to create a table of estimated values of Compare the estimated values with the actual data (d) Use the model to estimate the percent of females that had offspring when there were 7 females (e) Use the model to estimate the number of females when % of the females had offspring Ω 7 volts 8 volts Advanced Applications In Eercises 7 78, find values of,, and that satisf the sstem These sstems arise in certain optimization problems, and is called a Lagrange multiplier True or False? In Eercises 79 8, determine whether the statement is true or false If it is false, eplain wh or give an eample that shows it is false 79 The sstem z z z is in row-echelon form 8 If a sstem of three linear equations is inconsistent, then its graph has no points common to all three equations 8 The sstem z z z is consistent and has a unique solution In Eercises 8 8, find two sstems of linear equations that have the ordered triple as a solution (There are man correct answers) 8,, 8,, 8,, 7 8,, 7 CAPSTONE 8 Find values of a, b, and c (if possible) such that the sstem of linear equations has (a) a unique solution, (b) no solution, and (c) an infinite number of solutions a b z z cz

26 88 Chapter Sstems of Equations and Matrices Sstems of Inequalities Sketch the graphs of inequalities in two variables Solve sstems of inequalities Use sstems of inequalities in two variables to model and solve real-life problems The Graph of an Inequalit The statements < and are inequalities in two variables An ordered pair a, b is a solution of an inequalit in and if the inequalit is true when a and b are substituted for and, respectivel The graph of an inequalit is the collection of all solutions of the inequalit To sketch the graph of an inequalit, begin b sketching the graph of the corresponding equation The graph of the equation will normall separate the plane into two or more regions In each such region, one of the following must be true All points in the region are solutions of the inequalit No point in the region is a solution of the inequalit So, ou can determine whether the points in an entire region satisf the inequalit b simpl testing one point in the region GUIDELINES FOR SKETCHING THE GRAPH OF AN INEQUALITY IN TWO VARIABLES Replace the inequalit sign b an equal sign, and sketch the graph of the resulting equation (Use a dashed line for < or > and a solid line for or ) Test one point in each of the regions formed b the graph in Step If the point satisfies the inequalit, shade the entire region to denote that ever point in the region satisfies the inequalit EXAMPLE Sketching the Graph of an Inequalit Sketch the graph of Solution Begin b graphing the corresponding equation, which is a parabola, as shown in Figure 8 B testing a point above the parabola, and a point below the parabola,, ou can see that the points that satisf the inequalit are those ling above (or on) the parabola = STUDY TIP Note that when sketching the graph of an inequalit in two variables, a dashed line means all points on the line or curve are not solutions of the inequalit A solid line means all points on the line or curve are solutions of the inequalit Figure 8 (, ) Test point above parabola Test point below parabola (, )