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All Textbook Solutions for Elementary Linear Algebra (MindTap Course List)

Linear Equations. In Exercises 1-6, determine whether the equation is linear in the variables x and y. 2x3y=4 Linear Equations. In Exercises 1-6, determine whether the equation is linear in the variables x and y. 3x4xy=0 Linear Equations. In Exercises 1-6, determine whether the equation is linear in the variables x and y. 3y+2x1=0 Linear Equations. In Exercises 1-6, determine whether the equation is linear in the variables x and y. x2+y2=4 Linear Equations. In Exercises 1-6, determine whether the equation is linear in the variables x and y. 2sinxy=14 Linear Equations. In Exercises 1-6, determine whether the equation is linear in the variables x and y. (cos3)x+y=16 Parametric Representation. In Exercises 7-10, find a parametric representation of the solution set of the linear equation. 2x4y=0 Parametric Representation. In Exercises 7-10, find a parametric representation of the solution set of the linear equation. 3x12y=9 Parametric Representation. In Exercises 7-10, find a parametric representation of the solution set of the linear equation. x+y+z=1 Parametric Representation. In Exercises 7-10, find a parametric representation of the solution set of the linear equation. 12x1+24x236x3=12 Graphing Analysis. In Exercises 11-24, graph the system of linear equations. Solve the system and interpret your answer. 2x+y=4xy=2 Graphing Analysis. In Exercises 11-24, graph the system of linear equations. Solve the system and interpret your answer. x+3y=2x+2y=3 Graphing Analysis. In Exercises 11-24, graph the system of linear equations. Solve the system and interpret your answer. x+y=13x3y=4 Graphing Analysis. In Exercises 11-24, graph the system of linear equations. Solve the system and interpret your answer. 12x13y=12x+43y=4 Graphing Analysis. In Exercises 11-24, graph the system of linear equations. Solve the system and interpret your answer. 3x5y=72x+y=9 Graphing Analysis. In Exercises 11-24, graph the system of linear equations. Solve the system and interpret your answer. x+3y=174x+3y=7 Graphing Analysis. In Exercises 11-24, graph the system of linear equations. Solve the system and interpret your answer. 2xy=55xy=11 Graphing Analysis. In Exercises 11-24, graph the system of linear equations. Solve the system and interpret your answer. x5y=216x+5y=21 Graphing Analysis. In Exercises 11-24, graph the system of linear equations. Solve the system and interpret your answer. x+34+y13=12xy=12 Graphing Analysis. In Exercises 11-24, graph the system of linear equations. Solve the system and interpret your answer. x12+y+23=4x2y=5 Graphing Analysis. In Exercises 11-24, graph the system of linear equations. Solve the system and interpret your answer. 0.05x0.03y=0.070.07x+0.02y=0.16 22E 23E Graphing Analysis. In Exercises 11-24, graph the system of linear equations. Solve the system and interpret your answer. 2x3+y6=234x+y=4 Back-Substitution. In Exercises 25-30, use back-substitution to solve the system. x1x2=2x2=3 Back-Substitution. In Exercises 25-30, use back-substitution to solve the system. 2x14x2=63x2=9 Back-Substitution. In Exercises 25-30, use back-substitution to solve the system. x+yz=02y+z=312z=0 Back-Substitution. In Exercises 25-30, use back-substitution to solve the system. xy=53y+z=114z=8 Back-Substitution. In Exercises 25-30, use back-substitution to solve the system. 5x1+2x2+x3=02x1+x2=0 Back-Substitution. In Exercises 25-30, use back-substitution to solve the system. x1+x2+x3=0x2=0 31E Graphical Analysis. In Exercises 31-36, complete parts a-e for the system of equations. a Use a graphing utility to graph the system. b Use the graph to determine whether the system is consistent or inconsistent. c If the system is consistent, approximate the solution. d Solve the system algebraically. e Compare the solution in part d with the approximation in part c. What can you conclude? 4x5y=38x+10y=14 Graphical Analysis. In Exercises 31-36, complete parts a-e for the system of equations. a Use a graphing utility to graph the system. b Use the graph to determine whether the system is consistent or inconsistent. c If the system is consistent, approximate the solution. d Solve the system algebraically. e Compare the solution in part d with the approximation in part c. What can you conclude? 2x8y=312x+y=0 34E 35E Graphical Analysis. In Exercises 31-36, complete parts a-e for the system of equations. a Use a graphing utility to graph the system. b Use the graph to determine whether the system is consistent or inconsistent. c If the system is consistent, approximate the solution. d Solve the system algebraically. e Compare the solution in part d with the approximation in part c. What can you conclude? 14.7x+2.1y=1.0544.1x6.3y=3.15 System of Linear Equations. In Exercises 37-56, solve the system of linear equations. x1x2=03x12x2=1 38E System of Linear Equations. In Exercises 37-56, solve the system of linear equations. 3u+v=240u+3v=240 40E System of Linear Equations. In Exercises 37-56, solve the system of linear equations. 9x3y=115x+25y=13 System of Linear Equations. In Exercises 37-56, solve the system of linear equations. 23x1+16x2=04x1+x2=0 43E System of Linear Equations. In Exercises 37-56, solve the system of linear equations. x1+43+x2+12=13x1x2=2 System of Linear Equations. In Exercises 37-56, solve the system of linear equations. 0.02x10.05x2=0.190.03x1+0.04x2=0.52 System of Linear Equations. In Exercises 37-56, solve the system of linear equations. 0.05x10.03x2=0.210.07x1+0.02x2=0.17 System of Linear Equations. In Exercises 37-56, solve the system of linear equations. xyz=0x+2yz=62xz=5 System of Linear Equations. In Exercises 37-56, solve the system of linear equations. x+y+z=2x+3y+2z=84x+y=4 System of Linear Equations. In Exercises 37-56, solve the system of linear equations. 3x12x2+4x3=1x1+x22x3=32x13x2+6x3=8 System of Linear Equations. In Exercises 37-56, solve the system of linear equations. 5x13x2+2x3=32x1+4x2x3=7x111x2+4x3=3 System of Linear Equations. In Exercises 37-56, solve the system of linear equations. 2x1+x23x3=44x1+2x3=102x1+3x213x3=8 System of Linear Equations. In Exercises 37-56, solve the system of linear equations. x1+4x3=134x12x2+x3=72x12x27x3=19 System of Linear Equations. In Exercises 37-56, solve the system of linear equations. x3y+2z=185x15y+10z=18 System of Linear Equations. In Exercises 37-56, solve the system of linear equations. x12x2+5x3=23x1+2x2x3=2 System of Linear Equations. In Exercises 37-56, solve the system of linear equations. x+y+z+w=62x+3yw=03x+4y+z+2w=4x+2yz+w=0 System of Linear Equations. In Exercises 37-56, solve the system of linear equations. x1+2x4=14x2x3x4=2x2x4=03x12x2+3x3=4 System of Linear Equations. In Exercises 57-62, use a software program or a graphing utility to solve the system of linear equations. 123.5x+61.3y32.4z=262.7454.7x45.6y+98.2z=197.442.4x89.3y+12.9z=33.66 System of Linear Equations. In Exercises 57-62, use a software program or a graphing utility to solve the system of linear equations. 120.2x+62.4y36.5z=258.6456.8x42.8y+27.3z=71.4488.1x+72.5y28.5z=225.88 System of Linear Equations. In Exercises 57-62, use a software program or a graphing utility to solve the system of linear equations. x1+0.5x2+0.33x3+0.25x4=1.10.5x1+0.33x2+0.25x3+0.21x4=1.20.33x1+0.25x2+0.2x3+0.17x4=1.30.25x1+0.2x2+0.17x3+0.14x4=1.4 System of Linear Equations. In Exercises 57-62, use a software program or a graphing utility to solve the system of linear equations. 0.1x2.5y+1.2z0.75w=1082.4x+1.5y1.8z+0.25w=810.4x3.2y+1.6z1.4w=148.81.6x+1.2y3.2z+0.6w=143.2 System of Linear Equations. In Exercises 57-62, use a software program or a graphing utility to solve the system of linear equations. 12x137x2+29x3=34963023x1+49x225x3=194545x118x2+43x3=139150 System of Linear Equations. In Exercises 57-62, use a software program or a graphing utility to solve the system of linear equations. 18x17y+16z15w=117x+16y15z+14w=116x15y+14z13w=115x+14y13z+12w=1 Number of Solutions. In Exercises 63-66, state why the system of equations must have at least one solution. Then solve the system and determine whether it has exactly one solution or infinitely many solutions. 4x+3y+17z=05x+4y+22z=04x+2y+19z=0 Number of Solutions. In Exercises 63-66, state why the system of equations must have at least one solution. Then solve the system and determine whether it has exactly one solution or infinitely many solutions. 2x+3y=04x+3yz=08x+3y+3z=0 Number of Solutions. In Exercises 63-66, state why the system of equations must have at least one solution. Then solve the system and determine whether it has exactly one solution or infinitely many solutions. 5x+5yz=010x+5y+2z=05x+15y9z=0 Number of Solutions. In Exercises 63-66, state why the system of equations must have at least one solution. Then solve the system and determine whether it has exactly one solution or infinitely many solutions. 16x+3y+z=016x+2yz=0 Nutrition One eight-ounce glass of apple juice and one eight-ounce glass of orange juice contain a total of 227 milligrams of vitamin C. Two eight-ounce glasses of apple juice and three eight-ounce glasses of orange juice contain a total of 578 milligrams of vitamin C. How much vitamin C is in an eight-ounce glass of each type of juice?Airplane Speed Two planes start from Los Angeles International Airport and fly in opposite directions. The second plane starts 12 hour after the first plane, but its speed is 80 kilometers per hour faster. Two hours after the first plane departs, the planes are 3200 kilometers apart. Find the airspeed of each plane.True or False? In Exercises 69 and 70, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a A system of one linear equation in two variables is always consistent. b A system of two linear equations in three variables is always consistent. c If a linear system is consistent, then it has infinitely many solutions.True or False? In Exercises 69 and 70, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a A linear system can have exactly two solutions. b Two systems of linear equations are equivalent when they have the same solution set. c A system of three linear equations in two variables is always inconsistent.Find a system of two equations in two variables, x1 and x2, that has the solution set given by the parametric representation x1=t and x2=3t4, where t is any real number. Then show that the solutions to the system can also be written as x1=43+t3 and x2=t.Find a system of two equations in three variables, x1, x2 and x3 that has the solution set given by the parametric representation x1=t, x2=s and x3=3+st, where s and t are any real numbers. Then show that the solutions to the system can also be written as x1=3+st,x2=s and x3=t.Substitution In Exercises 73-76, solve the system of equations by first letting A=1x,B=1y, C=1z. 12x12y=73x+4y=0 Substitution In Exercises 73-76, solve the system of equations by first letting A=1x,B=1y, C=1z. 3x+2y=12x3y=176 75E Substitution In Exercises 73-76, solve the system of equations by first letting A=1x,B=1y, C=1z. 2x+1y2z=53x4y=12x+1y+3z=0 77E Trigonometric Coefficients In Exercises 77 and 78, solve the system of linear equations for x and y. (cos)x+(sin)y=1(sin)x+(cos)y=1 Coefficient Design In Exercises 79-84, determine the values of k such that the system of linear equations has the indicated number of solutions. No solution x+ky=2kx+y=4 Coefficient Design In Exercises 79-84, determine the values of k such that the system of linear equations has the indicated number of solutions. Exactly one solution x+ky=0kx+y=0 Coefficient Design In Exercises 79-84, determine the values of k such that the system of linear equations has the indicated number of solutions. Exactly one solution kx+2ky+3kz=4kx+y+z=02xy+z=1 Coefficient Design In Exercises 79-84, determine the values of k such that the system of linear equations has the indicated number of solutions. No solution x+2y+kz=63x+6y+8z=4 83E Coefficient Design In Exercises 79-84, determine the values of k such that the system of linear equations has the indicated number of solutions. Infinitely many solutions kx+y=163x4y=64 Determine the values of k such that the system of linear equations does not have a unique solution. x+y+kz=3x+ky+z=2kx+y+z=1 CAPSTONE Find values of a, b, and c such that the system of linear equations has a exactly one solution, b infinitely many solutions, and c no solution. Explain. x+5y+z=0x+6yz=02x+ay+bz=c Writing Consider the system of linear equations in x and y. a1x+b1y=c1a2x+b2y=c2a3x+b3y=c3 Describe the graphs of these three equations in the xy-plane when the system has a exactly one solution, b infinitely many solutions, and c no solution.Writing Explain why the system of linear equations in Exercise 87 must be consistent when the constant terms c1,c2,c3 are all zero. Reference: 87. Writing Consider the system of linear equations in x and y. a1x+b1y=c1a2x+b2y=c2a3x+b3y=c3 Describe the graphs of these three equations in the xy-plane when the system has a exactly one solution, b infinitely many solutions, and c no solution.Show that if ax2+bx+c=0 for all x, then a=b=c=0.Consider the system of linear equations in x and y. ax+by=ecx+dy=f Under what conditions will the system have exactly one solution?Discovery In Exercises 91 and 92, sketch the lines represented by the system of equations. Then use Gaussian elimination to solve the system. At each step of the elimination process, sketch the corresponding lines. What do you observe about the lines? x4y=35x6y=13 Discovery In Exercises 91 and 92, sketch the lines represented by the system of equations. Then use Gaussian elimination to solve the system. At each step of the elimination process, sketch the corresponding lines. What do you observe about the lines? 2x3y=74x+6y=14 93E Writing In Exercises 93 and 94, the graphs of the two equations appear to be parallel. Solve the system of equations algebraically. Explain why the graphs are misleading. 21x20y=013x12y=120 Matrix sizeIn Exercises 1-6, determine the size of the matrix. [124346012]Matrix sizeIn Exercises 1-6, determine the size of the matrix. [2112]Matrix sizeIn Exercises 1-6, determine the size of the matrix. [21621101]4E Matrix sizeIn Exercises 1-6, determine the size of the matrix. [86412174111211203110]6E Elementary Row Operations In Exercises 7-10, identify the elementary row operations being performed to obtain the new row-equivalent matrix. OriginalMatrixNewRowEquivalentMatrix[251318][13039318]Elementary Row Operations In Exercises 7-10, identify the elementary row operations being performed to obtain the new row-equivalent matrix. OriginalMatrixNewRowEquivalentMatrix[314437][314505]Elementary Row Operations In Exercises 7-10, identify the elementary row operations being performed to obtain the new row-equivalent matrix. OriginalMatrixNewRowEquivalentMatrix[017715873212][158701770132323]Elementary Row Operations In Exercises 7-10, identify the elementary row operations being performed to obtain the new row-equivalent matrix. OriginalMatrixNewRowEquivalentMatrix[123225175476][1232097110684]Augmented Matrix In Exercises 11-18, find the solution set of the system of linear equations represented by the augmented matrix. [100012]Augmented Matrix In Exercises 11-18, find the solution set of the system of linear equations represented by the augmented matrix. [102013]Augmented Matrix In Exercises 11-18, find the solution set of the system of linear equations represented by the augmented matrix. [110301210011]Augmented Matrix In Exercises 11-18, find the solution set of the system of linear equations represented by the augmented matrix. [121000110000]Augmented Matrix In Exercises 11-18, find the solution set of the system of linear equations represented by the augmented matrix. [211311100121]Augmented Matrix In Exercises 11-18, find the solution set of the system of linear equations represented by the augmented matrix. [311512101012]Augmented Matrix In Exercises 11-18, find the solution set of the system of linear equations represented by the augmented matrix. [12014012130012100014]Augmented Matrix In Exercises 11-18, find the solution set of the system of linear equations represented by the augmented matrix. [12013013010012000002]Row-Echelon Form In Exercises 19-24, determine whether the matrix is in row-echelon form. If it is, determine whether it is also in reduced row-echelon form. [100011000020]Row-Echelon Form In Exercises 19-24, determine whether the matrix is in row-echelon form. If it is, determine whether it is also in reduced row-echelon form. [01100021]Row-Echelon Form In Exercises 19-24, determine whether the matrix is in row-echelon form. If it is, determine whether it is also in reduced row-echelon form. [201012000512]Row-Echelon Form In Exercises 19-24, determine whether the matrix is in row-echelon form. If it is, determine whether it is also in reduced row-echelon form. [102013001140]Row-Echelon Form In Exercises 19-24, determine whether the matrix is in row-echelon form. If it is, determine whether it is also in reduced row-echelon form. [001000000012000]Row-Echelon Form In Exercises 19-24, determine whether the matrix is in row-echelon form. If it is, determine whether it is also in reduced row-echelon form. [100000000010]System of Linear Equations In Exercises 25-38, solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. x+3y=113x+y=9 System of Linear Equations In Exercises 25-38, solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. 2x+6y=162x6y=16 System of Linear Equations In Exercises 25-38, solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. x+2y=1.52x4y=3 System of Linear Equations In Exercises 25-38, solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. 2xy=0.13x+2y=1.6 System of Linear Equations In Exercises 25-38, solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. 3x+5y=223x+4y=44x8y=32 System of Linear Equations In Exercises 25-38, solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. x+2y=0x+y=63x2y=8 System of Linear Equations In Exercises 25-38, solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. x13x3=23x1+x22x3=52x1+2x2+x3=4 System of Linear Equations In Exercises 25-38, solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. 3x12x2+3x3=223x2x3=246x17x2=22 System of Linear Equations In Exercises 25-38, solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. 2x1+3x3=34x13x2+7x3=58x19x2+15x3=10 System of Linear Equations In Exercises 25-38, solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. x1+x25x3=3x12x3=12x1x2x3=0 System of Linear Equations In Exercises 25-38, solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. 4x+12y7z20w=223x+9y5z28w=30 36E System of Linear Equations In Exercises 25-38, solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. 3x+3y+12z=6x+y+4z=22x+5y+20z=10x+2y+8z=4 System of Linear Equations In Exercises 25-38, solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. 2x+yz+2w=63x+4y+w=1x+5y+2z+6w=35x+2yzw=3 39E 40E 41E System of Linear Equations In Exercises 39-42, use a software program or a graphing utility to solve the system of linear equations. x1+2x22x3+2x4x5+3x6=02x1x2+3x3+x43x5+2x6=17x1+3x22x3+x42x53x6=53x12x2+x3x4+3x52x6=1x12x2+x3+3x42x5+x6=10x13x2+x3+3x42x5+x6=11 Homogeneous System In Exercises 43-46, solve the homogeneous linear system corresponding to the given coefficient matrix. [100011000]Homogeneous System In Exercises 43-46, solve the homogeneous linear system corresponding to the given coefficient matrix. [10010100]Homogeneous System In Exercises 43-46, solve the homogeneous linear system corresponding to the given coefficient matrix. [100000010100]Homogeneous System In Exercises 43-46, solve the homogeneous linear system corresponding to the given coefficient matrix. [000000000]47E 48E Matrix Representation In Exercises 49 and 50, assume that the matrix is the augmented matrix of a system of linear equations, and a determine the number of equations and the number of variables, and b find the values of k such that the system is consistent. Then assume that the matrix is the coefficient matrix of a homogeneous system of linear equations, and repeat parts a and b. A=[13k421]Matrix Representation In Exercises 49 and 50, assume that the matrix is the augmented matrix of a system of linear equations, and a determine the number of equations and the number of variables, and b find the values of k such that the system is consistent. Then assume that the matrix is the coefficient matrix of a homogeneous system of linear equations, and repeat parts a and b. A=[21342k426]Coefficient Design In Exercises 51 and 52, find values of a, b, and c if possible such that the system of linear equation has a a unique solution, b no solution, and c infinitely many solutions. x+y=2y+z=2x+z=2ax+by+cz=0 Coefficient Design In Exercises 51 and 52, find values of a, b, and c if possible such that the system of linear equation has a a unique solution, b no solution, and c infinitely many solutions. x+y=0y+z=0x+z=0ax+by+cz=0 53E 54E 55E 56E 57E 58E True or False? In Exercises 59 and 60, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) A 63 matrix has six rows. (b) Every matrix is row-equivalent to a matrix in row- echelon form. (c) If the row- echelon form of the augmented matrix of a system of linear equations contains the row 1 0 0 0 0, then the original system is inconsistent. (d) A homogeneous system of four linear equations in six variables has infinitely many solutions.60E 61E 62E 63E Row Equivalence In Exercises 63 and 64, determine conditions on a, b, c, and d such that the matrix [abcd] will be row-equivalent to the given matrix. [1000]Homogeneous System In Exercises 65 and 66, find all values of the Greek letter lambda for which the homogeneous linear system has nontrivial solutions. (2)x+y=0x+(2)y=0 66E 67E CAPSTONE In your own words, describe the difference between a matrix in row-echelon form and a matrix in reduced row echelon form. Include an example of each to support your explanation.69E 70E Polynomial Curve Fitting In Exercises 1-12,a determine the polynomial function whose graph passes through the points, and b sketch the graph of the polynomial function, showing the points. (2,5),(3,2),(4,5)Polynomial Curve Fitting In Exercises 1-12,a determine the polynomial function whose graph passes through the points, and b sketch the graph of the polynomial function, showing the points. (0,0),(2,2),(4,0)3E Polynomial Curve Fitting In Exercises 1-12,a determine the polynomial function whose graph passes through the points, and b sketch the graph of the polynomial function, showing the points. (2,4),(3,4),(4,4)Polynomial Curve Fitting In Exercises 1-12,a determine the polynomial function whose graph passes through the points, and b sketch the graph of the polynomial function, showing the points. (1,3),(0,0),(1,1)(4,58)6E 7E 8E Polynomial Curve Fitting In Exercises 1-12,a determine the polynomial function whose graph passes through the points, and b sketch the graph of the polynomial function, showing the points. (2013,5),(2014,7),(2015,12)10E 11E Polynomial Curve Fitting In Exercises 1-12,a determine the polynomial function whose graph passes through the points, and b sketch the graph of the polynomial function, showing the points. (1,1),(1.189,1.587),(1.316,2.080),(1.414,2.520)Use sin0=0, sin2=1, and sin=0 to estimate sin3.Use log21=0,log22=1, and log24=2 to estimate log23.15E 16E 17E Population The table shows the U.S. populations for the years 1970,1980,1990 and 2000. Source: U.S. Census Bureau Year Population in millions 1970 1980 1990 2000 205 227 249 282 a Find a cubic polynomial that fits the data and use it to estimate the population in 2010. b The actual population in 2010 was 309 million. How does your estimate compare?19E 20E Network Analysis The figure shows the flow of traffic in vehicles per hour through a network of streets. a Solve this system for xi,i=1,2,...,5. b Find the traffic flow when x3=0 and x5=100. c Find the traffic flow when x3=x5=100.Network Analysis The figure shows the flow of traffic in vehicles per hour through a network of streets. a Solve this system for xi,i=1,2,...,5. b Find the traffic flow when x2=200 and x3=50. c Find the traffic flow when x2=150 and x3=0.Network Analysis The figure shows the flow of traffic in vehicles per hour through a network of streets. a Solve this system for xi,i=1,2,3,4. b Find the traffic flow when x4=0. c Find the traffic flow when x4=100. d Find the traffic flow when x1=2x2.Network Analysis Water is flowing through a network of pipes in thousands of cubic meters per hour, as shown in figure. a Solve this system for the water flow represented by xii=1,2,...,7. b Find the water flow when x1=x2=100. c Find the water flow when x6=x7=0. d Find the water flow when x5=1000 and x6=0.25E 26E 27E 28E Temperature In Exercises 29 and 30, the figure shows the boundary temperatures in degree Celsius of an insulted thin metal plate. The steady-state temperature at an interior junction is approximately equal to the mean of the temperatures at the four surrounding junctions. Use a system of linear equations to approximate the interior temperatures T1,T2,T3, and T4.Temperature In Exercises 29 and 30, the figure shows the boundary temperatures in degree Celsius of an insulted thin metal plate. The steady-state temperature at an interior junction is approximately equal to the mean of the temperatures at the four surrounding junctions. Use a system of linear equations to approximate the interior temperatures T1,T2,T3, and T4.Partial Fraction Decomposition In Exercises 3134, use a system of equations to find the partial fraction decomposition of the rational expression. Solve the system using matrices. 4x2(x+1)2(x1)=Ax1+Bx+1+C(x+1)2 Partial Fraction Decomposition In Exercises 3134, use a system of equations to find the partial fraction decomposition of the rational expression. Solve the system using matrices. 3x27x12(x+4)(x4)2=Ax+4+Bx4+C(x4)2 33E 34E Calculus In Exercises 35 and 36, find the values of x,y, and that satisfy the system of equations. Such systems arise in certain problems of calculus, and is called the Lagrange multiplier. 2x+=02y+=0x+y4=0 36E Calculus The graph of a parabola passes through the points (0,1) and (12,12) and has a horizontal tangent line at (12,12). Find an equation for the parabola and sketch its graph.Calculus The graph of a cubic polynomial function has horizontal tangent lines at (1,2) and (1,2). Find an equation for the function and sketch its graph.39E 40E 41E 42E Linear Equations In Exercises 1-6, determine whether the equation is linear in the variables x and y. 2xy2=4 Linear Equations In Exercises 1-6, determine whether the equation is linear in the variables x and y. 2xy6y=0 3CR 4CR 5CR 6CR 7CR Parametric Representation In Exercises 7 and 8, find a parametric representation of the solution set of the linear equation. 3x1+2x24x3=0 9CR 10CR 11CR System of Linear Equations In Exercises 9-20, solve the system of linear equations. x=y+34x=y+10 13CR 14CR 15CR System of Linear Equations In Exercises 9-20, solve the system of linear equations. 40x1+30x2=2420x1+15x2=14 17CR 18CR 19CR 20CR 21CR 22CR 23CR 24CR 25CR 26CR 27CR 28CR 29CR 30CR 31CR 32CR 33CR 34CR 35CR System of Linear Equations In Exercises 31-40, solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. 2x+6z=93x2y+11z=163xy+7z=11 37CR System of Linear Equations In Exercises 31-40, solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. 2x1+5x219x3=343x1+8x231x3=54 39CR 40CR 41CR 42CR 43CR 44CR 45CR 46CR Homogeneous System In Exercises 47-50, solve the homogeneous system of linear equations. x12x28x3=03x1+2x2=0 48CR 49CR 50CR 51CR 52CR 53CR Find if possible values of a,b, and c such that the system of linear equations has a no solution, b exactly one solution, and c infinitely many solutions. 2xy+z=ax+y+2z=b3y+3z=c 55CR 56CR 57CR Find all values of for which the homogeneous system of linear equations has nontrivial solutions. (+2)x12x2+3x3=02x1+(1)x2+6x3=0x1+2x2+x3=0 59CR 60CR Sports In Super Bowl I, on January 15, 1967, the Green Bay Packers defeated the Kansas City Chiefs by a score of 35 to 10. The total points scored came from a combination of touchdowns, extra-point kicks, and field goals, worth 6,1, and 3 points, respectively. The numbers of touchdowns and extra-point kicks were equal. There were six times as many touchdowns as field goals. Find the numbers of touchdowns, extra-point kicks, and field goals scored. Source: National Football League Agriculture A mixture of 6 gallons of chemical A, 8 gallons of chemical B, and 13 gallons of chemical C is required to kill a destructive crop insect. Commercial spray X contains 1,2, and 2 parts, respectively, of these chemicals. Commercial spray Y contains only Chemical C. Commercial spray Z contains A, B, and C in equal amounts. How much of each type of commercial spray is needed to get the desired mixture?63CR 64CR 65CR 66CR 67CR 68CR 69CR 70CR 71CR Network Analysis Determine the currents I1,I2, and I3 for the electrical network shown in the figure.Equality of Matrices In Exercises 1-4, find x and y. x-27y=4-2722 Equality of Matrices In Exercises 1-4, find x and y. -5xy8=-513128 Equality of Matrices In Exercises 1-4, find x and y. 164531315024460=1642x+1-31315023y-543x0 Equality of Matrices In Exercises 1-4, find x and y. x+28-312y2x7-2y+2=2x+68-3118-87-211 Operations with Matrices In Exercises 5-10, find, if possible, a A+B, b A-B, c 2A, d 2A-B, and e B+12A. A=1221,B=-3-242 Operations with Matrices In Exercises 5-10, find, if possible, a A+B, b A-B, c 2A, d 2A-B, and e B+12A. A=6-124-35,B=14-15110 Operations with Matrices In Exercises 5-10, find, if possible, a A+B, b A-B, c 2A, d 2A-B, and e B+12A. A=211-1-14,B=2-34-31-2 Operations with Matrices In Exercises 5-10, find, if possible, a A+B, b A-B, c 2A, d 2A-B, and e B+12A. A=32-1245012,B=021542210 Operations with Matrices In Exercises 5-10, find, if possible, a A+B, b A-B, c 2A, d 2A-B, and e B+12A. A=603-1-40,B=8-14-3 Operations with Matrices In Exercises 5-10, find, if possible, a A+B, b A-B, c 2A, d 2A-B, and e B+12A. A=32-1,B=-462 Find a c21 and b c13, where C=2A-3B, A=544-312, and B=12-70-51 Find a c23 and b c32, where C=5A+2B, A=411-9032-311, and B=105-4611-649 Solve for x,y and z in the matrix equation 4xyz-1=2yz-x1+24x5-x.Solve for x,y,z and w in the matrix equation wxyx=-432-1+2ywzx.15E 16E Finding Products of Two Matrices In Exercises 15-28, find, if possible, a AB and b BA. A=2-1351-2223,B=012-413-4-1-2 Finding Products of Two Matrices In Exercises 15-28, find, if possible, a AB and b BA. A=1-172-1831-1,B=1122111-32 Finding Products of Two Matrices In Exercises 15-28, find, if possible, a AB and b BA. A=21-3416,B=0-104028-17 Finding Products of Two Matrices In Exercises 15-28, find, if possible, a AB and b BA. A=321-3044-2-4,B=122-11-2 21E 22E

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