Solution: The sketch a graph for each equation to determine

Problem 42E Solve the system of two nonlinear equations in two unknowns x12 – 2x1 + x22 = 3 x12 - x22 =

Problem 1E Which of the equation are linear? x1 + 2x3 = 3

Problem 2E Which of the equation are linear? x1x2 + x2 = 1

Problem 3E Which of the equation are linear? x1 – x2 = sin2 x1 + cos2 x1

Problem 5E Which of the equation are linear? |x1| - |x2| = 0

Problem 6E Which of the equation are linear?

Problem 7E The coefficients are given for a system of the form (2). Display the system and verify that the given values constitute a solution. a11 =0, a12 = 3. a13 = 4, a22 = 1, b1 = 7, b2 = 2, x1 = 1, x2 = 2

Problem 8E The coefficients are given for a system of the form (2). Display the system and verify that the given values constitute a solution. a11 =6, a12 = -1. a13 = 1, a21 = 1, a22 = 2, a23 = 1, b1 = 14, b2 = 4, x1 = 2, x2 = -1, x3 = 1

Problem 9E The coefficients are given for a system of the form (2). Display the system and verify that the given values constitute a solution. a11 =1, a12 = 1, a21 = 1, a22 = 4, a31 = -1, a32 = 2, b1 = 0, b2 = -1, b2 = -3, x1 = 1, x2 = -1

Problem 10E The coefficients are given for a system of the form (2). Display the system and verify that the given values constitute a solution. a11 =0, a12 = 3, a21 = 4, a22 = 0, b1 = 9, b2 = 8, x1 = 2, x2 = 3

Problem 11E The sketch a graph for each equation to determine whether the system has a unique solution, no solution, or infinitely many solutions. 2x + y = 5 x – y = 1

Problem 12E The sketch a graph for each equation to determine whether the system has a unique solution, no solution, or infinitely many solutions 2x - y = -1 2x – y = 2

Problem 13E The sketch a graph for each equation to determine whether the system has a unique solution, no solution, or infinitely many solutions 3x + 2y = 6 -6x-4y= -12

Problem 14E The sketch a graph for each equation to determine whether the system has a unique solution, no solution, or infinitely many solutions 2x -+y = 5 x – y = 1 x + 3y = 9

Problem 15E The (2 × 3) system of linear equations a1x + b1y +c1z = d1 a2x + b2y + c2z = d2 is represented geometrically by two planes. How are the planes related when: a) The system has no solution? ________________ b) The system has infinitely many solutions? Is it possible for the system to have a unique solution? Explain.

Problem 16E Determine whether the given (2 × 3) system of linear equations represents coincident planes (that is, the same plane), two parallel planes, or two planes whose intersection is a line. In the latter case, give the parametric equations for the line; that is, give equations of the form x = at + b, y = ct + d, z = et + f. 2x1 + x2 + x3 = 3 -2x1 + x2 –x3 = 1

Problem 17E Determine whether the given (2 × 3) system of linear equations represents coincident planes (that is, the same plane), two parallel planes, or two planes whose intersection is a line. In the latter case, give the parametric equations for the line; that is, give equations of the form x = at + b, y = ct + d, z = et + f. x1 + 2x2 – x3 = 2 x1 + x2 + x3 = 3

Problem 19E Display the (2 × 3) matrix A= (aij), where a11 =2, a12= 1, a13 = 6, a21 = 4, a22 = 3, and a23 = 8.

Problem 18E Determine whether the given (2 × 3) system of linear equations represents coincident planes (that is, the same plane), two parallel planes, or two planes whose intersection is a line. In the latter case, give the parametric equations for the line; that is, give equations of the form x = at + b, y = ct + d, z = et + f. x1 + 3x2 – 2x3 = -1 2x1 + 6x2 – 4x3 = -2

Problem 20E Display the (2 × 4) matrix C = (cij), where ?23 = 4, c12 = 2, c21 = 2, c14 = 1, c22 = 2, c24 = 3, c11= 1, and c13 = 7

Problem 21E Display the (3 × 3) matrix Q = (qij), where q23 = 1, q32 = 2, q11 = 1, q13 = -3, q22 = 1, q33 = 1, q21 = 2, q12 = 4, and q31 = 3.

Problem 22E Suppose the matrix C Display the (2 × 4) matrix C = (cij), where ?23 = 4, c12 = 2, c21 = 2, c14 = 1, c22 = 2, c24 = 3, c11= 1, and c13 = 7 is the augmented matrix for a system of linear equations. Display the system.

Problem 23E Suppose the matrix C Display the (2 × 4) matrix C = (cij), where ?23 = 4, c12 = 2, c21 = 2, c14 = 1, c22 = 2, c24 = 3, c11= 1, and c13 = 7 is the augmented matrix for a system of linear equations. Display the system.for the matrices in Display the (2 x 3) matrix A= (aij), where a11=2, a12= 1, a13 = 6, a21 = 4, a22 = 3, and a23 = 8. and Display the (3 × 3) matrix Q = (qij), where q23 = 1, q32 = 2, q11 = 1, q13 = -3, q22 = 1, q33 = 1, q21 = 2, q12 = 4, and q31 = 3. display the coefficient matrix A and the augmented matrix B for the given system. x1 – x2 = -1 x1+ x2 = 3

Problem 25E Suppose the matrix C Display the (2 × 4) matrix C = (cij), where ?23 = 4, c12 = 2, c21 = 2, c14 = 1, c22 = 2, c24 = 3, c11= 1, and c13 = 7 is the augmented matrix for a system of linear equations. Display the system.for the matrices in Display the (2 x 3) matrix A= (aij), where a11=2, a12= 1, a13 = 6, a21 = 4, a22 = 3, and a23 = 8. and Display the (3 × 3) matrix Q = (qij), where q23 = 1, q32 = 2, q11 = 1, q13 = -3, q22 = 1, q33 = 1, q21 = 2, q12 = 4, and q31 = 3. Display the coefficient matrix A and the augmented matrix B for the given system. x1 + 3x2 – x3= 1 2x1 + 5x2 + x3 =5 x1 + x2 + x3 = 3

Problem 24E Suppose the matrix C Display the (2 × 4) matrix C = (cij), where ?23 = 4, c12 = 2, c21 = 2, c14 = 1, c22 = 2, c24 = 3, c11= 1, and c13 = 7 is the augmented matrix for a system of linear equations. Display the system.for the matrices in Display the (2 x 3) matrix A= (aij), where a11=2, a12= 1, a13 = 6, a21 = 4, a22 = 3, and a23 = 8. and Display the (3 × 3) matrix Q = (qij), where q23 = 1, q32 = 2, q11 = 1, q13 = -3, q22 = 1, q33 = 1, q21 = 2, q12 = 4, and q31 = 3. display the coefficient matrix A and the augmented matrix B for the given system. x1+ x2 – x3 = 2 2x1 – x3 = 1

Problem 26E Suppose the matrix C Display the (2 × 4) matrix C = (cij), where ?23 = 4, c12 = 2, c21 = 2, c14 = 1, c22 = 2, c24 = 3, c11= 1, and c13 = 7 is the augmented matrix for a system of linear equations. Display the system.for the matrices in Display the (2 x 3) matrix A= (aij), where a11=2, a12= 1, a13 = 6, a21 = 4, a22 = 3, and a23 = 8. and Display the (3 × 3) matrix Q = (qij), where q23 = 1, q32 = 2, q11 = 1, q13 = -3, q22 = 1, q33 = 1, q21 = 2, q12 = 4, and q31 = 3. Display the coefficient matrix A and the augmented matrix B for the given system x1 + x2 + 2x3 = 6 3x1 + 4x2 – x3 = 5 – x1 + x2 + x3 = 2

Problem 27E Suppose the matrix C Display the (2 × 4) matrix C = (cij), where ?23 = 4, c12 = 2, c21 = 2, c14 = 1, c22 = 2, c24 = 3, c11= 1, and c13 = 7 is the augmented matrix for a system of linear equations. Display the system.for the matrices in Display the (2 x 3) matrix A= (aij), where a11=2, a12= 1, a13 = 6, a21 = 4, a22 = 3, and a23 = 8. and Display the (3 × 3) matrix Q = (qij), where q23 = 1, q32 = 2, q11 = 1, q13 = -3, q22 = 1, q33 = 1, q21 = 2, q12 = 4, and q31 = 3. Display the coefficient matrix A and the augmented matrix B for the given system x1+ x2 – 3x3 = -1 x1+ 2x2 – 5x3 = -2 -x1? 3x2 + 7x3 = 3

Problem 28E Suppose the matrix C Display the (2 × 4) matrix C = (cij), where ?23 = 4, c12 = 2, c21 = 2, c14 = 1, c22 = 2, c24 = 3, c11= 1, and c13 = 7 is the augmented matrix for a system of linear equations. Display the system.for the matrices in Display the (2 x 3) matrix A= (aij), where a11=2, a12= 1, a13 = 6, a21 = 4, a22 = 3, and a23 = 8. and Display the (3 × 3) matrix Q = (qij), where q23 = 1, q32 = 2, q11 = 1, q13 = -3, q22 = 1, q33 = 1, q21 = 2, q12 = 4, and q31 = 3. Display the coefficient matrix A and the augmented matrix B for the given system x1 + x2 + x3 = 1 2x1 + 3x2 + x3 = 2, x1 - x2 + 3x3 = 2

Problem 29E Display the augmented matrix for the given system. Use elementary operations on equations to obtain an equivalent system of equations in which x1 appears in the first equation with coefficient one and has been eliminated from the remaining equations. Simultaneously, perform the corresponding elementary row operations on the augmented matrix. 2x1 + 3x2 = 6 4x1 – x2 = 7

Problem 30E Display the augmented matrix for the given system. Use elementary operations on equations to obtain an equivalent system of equations in which x1 appears in the first equation with coefficient one and has been eliminated from the remaining equations. Simultaneously, perform the corresponding elementary row operations on the augmented matrix. x1 + 2x2 – x3 = 1 x1 + x2 + 2x3 = 2 -2x1+ x2 = 4

Problem 31E Display the augmented matrix for the given system. Use elementary operations on equations to obtain an equivalent system of equations in which x1 appears in the first equation with coefficient one and has been eliminated from the remaining equations. Simultaneously, perform the corresponding elementary row operations on the augmented matrix x2 + x3 =4 x1 – x2 +2x3 =1 2x1 + x2- x3 = 0

Problem 32E Display the augmented matrix for the given system. Use elementary operations on equations to obtain an equivalent system of equations in which x1 appears in the first equation with coefficient one and has been eliminated from the remaining equations. Simultaneously, perform the corresponding elementary row operations on the augmented matrix x1 + x2 = 9 x1 – x2 =7 3x1 + x2 = 6

Problem 33E Display the augmented matrix for the given system. Use elementary operations on equations to obtain an equivalent system of equations in which x1 appears in the first equation with coefficient one and has been eliminated from the remaining equations. Simultaneously, perform the corresponding elementary row operations on the augmented matrix x1 + x2 + x3 + -x4 = 1 -x1+2x2 –x3 + x4 = 3 –x1 + x2 + 7x3 – x4 = 2

Problem 34E Display the augmented matrix for the given system. Use elementary operations on equations to obtain an equivalent system of equations in which x1 appears in the first equation with coefficient one and has been eliminated from the remaining equations. Simultaneously, perform the corresponding elementary row operations on the augmented matrix x2 + x3 – x4 = 3 x1 + 2x2 – x3 + x4 =1 –x1 + x2 + 7x3 – x4 = 0

Problem 35E Display the augmented matrix for the given system. Use elementary operations on equations to obtain an equivalent system of equations in which x1 appears in the first equation with coefficient one and has been eliminated from the remaining equations. Simultaneously, perform the corresponding elementary row operations on the augmented matrix x1 + x2 = 0 x1 – x2 = 0 3x1 + x2 = 0

Problem 36E Consider the equation 2x1 ? 3x2 + x3 - x4 = 3. a) In the six different possible combinations, set any two of the variables equal to 1 and graph the equation in terms of the other two. ________________ b) What type of graph do you always get when you set two of the variables equal to two fixed constants? ________________ c) What is one possible reason the equation in formula (1) is called linear?

Problem 38E In the following (2 × 2) linear systems (A) and (B), c is a nonzero scalar. Prove that any solution, x1 = s1, x2 = s2, for (A) is also a solution for (B). Conversely, show that any solution, x1 = t1

Problem 37E Consider the (2 × 2) system a11 + a12 x1 = b1 a21 x1 + a22 x2 = b2 Show that if a11a22 – a12a21 ? 0, then this system is equivalent to a system of the form c11x1 + c1 x2 = d1 c22 x2 = d2 where c11 ? 0 and c22 ? 0. Note that the second system always has a solution. [Hint: First suppose that a11 ? 0, and then consider the special case in which a11 = 0.]

Problem 39E In the following (2 × 2) linear systems (A) and (B), c is a nonzero scalar. Prove that any solution, x1 = s1, x2 = s2, for (A) is also a solution for (B). Conversely, show that any solution, x1 = t1

Problem 40E In the (2 × 2) linear systems that follow, the system (B) is obtained from (A) by performing the elementary operation E2 + cE1. Prove that any solution, x1 = s1, x2 = s2, for (A) is a solution for (B), Similarly, prove that any solution, x1 = t1, x2 = t2, for (B) is a solution for (A). (A)a11 + a12 x2 = b1 a21 x1 + a22 x2 = b2 ________________ (B) a11 + a12 x2 = b1 (a21+ca11)x11 + (a22 + ca12)x2 = b2 + cb1

Problem 41E Prove that any of the elementary operations in Theorem 1 applied to system (2) produces an equivalent system. [Hint: To simplify this proof, represent the ith equation in system (2) as = ; so for i = 1, 2, ... , m. With this notation, system (2) has the form of (A), which follows. Next, for exam-ple, if a multiple of c times the jth equation is added to the kth equation, a new system of the form (B) is produced: where = . To show that the operation gives an equivalent system, show that any solution for (A) is a solution for (B), and vice versa.] (Reference Theorem 1)

Problem 4E Which of the equation are linear? x1 – x2 = sin2 x1 + cos2 x2