Linear Algebra Ch 1.1 — Systems of Linear Equations

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Linear Algebra Ch 1.1 — Systems of Linear Equations 0/44
1 Systems of Linear Equations · Level 1
Solve each system by using elementary row operations on the equations or on the augmented matrix. \(x_1 + 5 x_2 = 7\) \(-2 x_1 - 7 x_2 = -5\)
2 Systems of Linear Equations · Level 1
Solve each system by using elementary row operations on the equations or on the augmented matrix. \(2 x_1 + 4 x_2 = -4\) \(5 x_1 + 7 x_2 = 11\)
3 Systems of Linear Equations · Level 1
Find the point \((x_1, x_2)\) that lies on the line \(x_1 + 5 x_2 = 7\) and on the line \(x_1 - 2 x_2 = -2\). See the figure.
4 Systems of Linear Equations · Level 1
Find the point of intersection of the lines \(x_1 - 5 x_2 = 1\) and \(3 x_1 - 7 x_2 = 5\).
5 Systems of Linear Equations · Level 2
Consider the matrix as the augmented matrix of a linear system. State in words the next two elementary row operations that should be performed in the process of solving the system. \(\begin{pmatrix} 1 & 3 & -4 & 0 & 9 \\ 1 & 1 & 5 & 0 & -8 \\ 0 & 0 & 1 & 0 & 7 \\ 0 & 0 & 0 & 1 & -6 \end{pmatrix}\)
6 Systems of Linear Equations · Level 2
Consider the matrix as the augmented matrix of a linear system. State in words the next two elementary row operations that should be performed in the process of solving the system. \(\begin{pmatrix} 1 & -6 & 4 & 0 & -1 \\ 0 & 2 & -7 & 0 & 4 \\ 0 & 0 & 1 & 2 & -3 \\ 0 & 0 & 3 & 1 & 6 \end{pmatrix}\)
7 Systems of Linear Equations · Level 2
The augmented matrix of a linear system has been reduced by row operations to the form shown. In each case, continue the appropriate row operations and describe the solution set of the original system. \(\begin{pmatrix} 1 & 7 & 3 & -4 \\ 0 & 1 & -1 & 3 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & -2 \end{pmatrix}\)
8 Systems of Linear Equations · Level 2
The augmented matrix of a linear system has been reduced by row operations to the form shown. In each case, continue the appropriate row operations and describe the solution set of the original system. \(\begin{pmatrix} 1 & 1 & 5 & 0 \\ 0 & 1 & 9 & 0 \\ 0 & 0 & 7 & -7 \end{pmatrix}\)
9 Systems of Linear Equations · Level 2
The augmented matrix of a linear system has been reduced by row operations to the form shown. In each case, continue the appropriate row operations and describe the solution set of the original system. \(\begin{pmatrix} 1 & -1 & 0 & 0 & -4 \\ 0 & 1 & -3 & 0 & -7 \\ 0 & 0 & 1 & -3 & -1 \\ 0 & 0 & 0 & 0 & 4 \end{pmatrix}\)
10 Systems of Linear Equations · Level 2
The augmented matrix of a linear system has been reduced by row operations to the form shown. In each case, continue the appropriate row operations and describe the solution set of the original system. \(\begin{pmatrix} 1 & -2 & 0 & 3 & 0 \\ 0 & 1 & 0 & -4 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{pmatrix}\)
11 Systems of Linear Equations · Level 2
Solve the system. \(x_2 + 4 x_3 = -4\) \(x_1 + 3 x_2 + 3 x_3 = -2\) \(3 x_1 + 7 x_2 + 5 x_3 = 6\)
12 Systems of Linear Equations · Level 2
Solve the system. \(x_1 - 3 x_2 + 4 x_3 = -4\) \(3 x_1 - 7 x_2 + 7 x_3 = -8\) \(-4 x_1 + 6 x_2 + 2 x_3 = 4\)
13 Systems of Linear Equations · Level 2
Solve the system. \(x_1 - 3 x_3 = 8\) \(2 x_1 + 2 x_2 + 9 x_3 = 7\) \(x_2 + 5 x_3 = -2\)
14 Systems of Linear Equations · Level 2
Solve the system. \(x_1 - 3 x_2 = 5\) \(-x_1 + x_2 + 5 x_3 = 2\) \(x_2 + x_3 = 0\)
15 Systems of Linear Equations · Level 2
Verify that the solution you found to Exercise 11 is correct by substituting the values you obtained back into the original equations.
16 Systems of Linear Equations · Level 2
Verify that the solution you found to Exercise 12 is correct by substituting the values you obtained back into the original equations.
17 Systems of Linear Equations · Level 2
Verify that the solution you found to Exercise 13 is correct by substituting the values you obtained back into the original equations.
18 Systems of Linear Equations · Level 2
Verify that the solution you found to Exercise 14 is correct by substituting the values you obtained back into the original equations.
19 Systems of Linear Equations · Level 3
Determine if the system is consistent. Do not completely solve the system. \(x_1 + 3 x_3 = 2\) \(x_2 - 3 x_4 = 3\) \(-2 x_2 + 3 x_3 + 2 x_4 = 1\) \(3 x_1 + 7 x_4 = -5\)
20 Systems of Linear Equations · Level 3
Determine if the system is consistent. Do not completely solve the system. \(x_1 - 2 x_4 = -3\) \(2 x_2 + 2 x_3 = 0\) \(x_3 + 3 x_4 = 1\) \(-2 x_1 + 3 x_2 + 2 x_3 + x_4 = 5\)
21 Systems of Linear Equations · Level 3
Do the three lines \(x_1 - 4 x_2 = 1\), \(2 x_1 - x_2 = -3\), and \(-x_1 - 3 x_2 = 4\) have a common point of intersection? Explain.
22 Systems of Linear Equations · Level 3
Do the three planes \(x_1 + 2 x_2 + x_3 = 4\), \(x_2 - x_3 = 1\), and \(x_1 + 3 x_2 = 0\) have at least one common point of intersection? Explain.
23 Systems of Linear Equations · Level 3
Determine the value(s) of \(h\) such that the matrix is the augmented matrix of a consistent linear system. \(\begin{pmatrix} 1 & h & 4 \\ 3 & 6 & 8 \end{pmatrix}\)
24 Systems of Linear Equations · Level 3
Determine the value(s) of \(h\) such that the matrix is the augmented matrix of a consistent linear system. \(\begin{pmatrix} 1 & h & -3 \\ -2 & 4 & 6 \end{pmatrix}\)
25 Systems of Linear Equations · Level 3
Determine the value(s) of \(h\) such that the matrix is the augmented matrix of a consistent linear system. \(\begin{pmatrix} 1 & 3 & -2 \\ -4 & h & 8 \end{pmatrix}\)
26 Systems of Linear Equations · Level 3
Determine the value(s) of \(h\) such that the matrix is the augmented matrix of a consistent linear system. \(\begin{pmatrix} 3 & -4 & h \\ -6 & 8 & 9 \end{pmatrix}\)
27 Systems of Linear Equations · Level 2
(T/F) Every elementary row operation is reversible.
28 Systems of Linear Equations · Level 2
(T/F) Elementary row operations on an augmented matrix never change the solution set of the associated linear system.
29 Systems of Linear Equations · Level 2
(T/F) A \(5 \times 6\) matrix has six rows.
30 Systems of Linear Equations · Level 2
(T/F) Two matrices are row equivalent if they have the same number of rows.
31 Systems of Linear Equations · Level 2
(T/F) The solution set of a linear system involving variables \(x_1, ..., x_n\) is a list of numbers \((s_1, ..., s_n)\) that makes each equation in the system a true statement when the values \(s_1, ..., s_n\) are substituted for \(x_1, ..., x_n\), respectively.
32 Systems of Linear Equations · Level 2
(T/F) An inconsistent system has more than one solution.
33 Systems of Linear Equations · Level 2
(T/F) Two fundamental questions about a linear system involve existence and uniqueness.
34 Systems of Linear Equations · Level 2
(T/F) Two linear systems are equivalent if they have the same solution set.
35 Systems of Linear Equations · Level 4
Find an equation involving \(g\), \(h\), and \(k\) that makes this augmented matrix correspond to a consistent system: \(\begin{pmatrix} 1 & -3 & 5 & g \\ 0 & 2 & -3 & h \\ -3 & 5 & -9 & k \end{pmatrix}\)
36 Systems of Linear Equations · Level 3
Construct three different augmented matrices for linear systems whose solution set is \(x_1 = -2, x_2 = 1, x_3 = 0\).
37 Systems of Linear Equations · Level 4
Suppose the system below is consistent for all possible values of \(f\) and \(g\). What can you say about the coefficients \(c\) and \(d\)? Justify your answer. \(x_1 + 5 x_2 = f\) \(c x_1 + d x_2 = g\)
38 Systems of Linear Equations · Level 4
Suppose \(a\), \(b\), \(c\), and \(d\) are constants such that \(a\) is not zero and the system below is consistent for all possible values of \(f\) and \(g\). What can you say about the numbers \(a\), \(b\), \(c\), and \(d\)? Justify your answer. \(a x_1 + b x_2 = f\) \(c x_1 + d x_2 = g\)
39 Systems of Linear Equations · Level 3
Find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first. \(\begin{pmatrix} 0 & -2 & 5 \\ 1 & 4 & -7 \\ 3 & -1 & 6 \end{pmatrix}\), \(\begin{pmatrix} 1 & 4 & -7 \\ 0 & -2 & 5 \\ 3 & -1 & 6 \end{pmatrix}\)
40 Systems of Linear Equations · Level 3
Find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first. \(\begin{pmatrix} 1 & 3 & -4 \\ 0 & -2 & 6 \\ 0 & -5 & 9 \end{pmatrix}\), \(\begin{pmatrix} 1 & 3 & -4 \\ 0 & 1 & -3 \\ 0 & -5 & 9 \end{pmatrix}\)
41 Systems of Linear Equations · Level 3
Find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first. \(\begin{pmatrix} 1 & -3 & 2 & 0 \\ 0 & 4 & -5 & 6 \\ 5 & -7 & 8 & -9 \end{pmatrix}\), \(\begin{pmatrix} 1 & -3 & 2 & 0 \\ 0 & 4 & -5 & 6 \\ 0 & 8 & -2 & -9 \end{pmatrix}\)
42 Systems of Linear Equations · Level 3
Find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first. \(\begin{pmatrix} 1 & 2 & -5 & 0 \\ 0 & 1 & -3 & -2 \\ 0 & -3 & 9 & 5 \end{pmatrix}\), \(\begin{pmatrix} 1 & 2 & -5 & 0 \\ 0 & 1 & -3 & -2 \\ 0 & 0 & 0 & -1 \end{pmatrix}\)
43 Systems of Linear Equations · Level 4
An important concern in the study of heat transfer is to determine the steady-state temperature distribution of a thin plate when the temperature around the boundary is known. Assume the plate shown in the figure represents a cross section of a metal beam, with negligible heat flow in the direction perpendicular to the plate. Let \(T_1, ..., T_4\) denote the temperatures at the four interior nodes of the mesh in the figure. The temperature at a node is approximately equal to the average of the four nearest nodes—to the left, above, to the right, and below. For instance, \(T_1 = (10 + 20 + T_2 + T_4) / 4\), or \(4 T_1 - T_2 - T_4 = 30\) Write a system of four equations whose solution gives estimates for the temperatures \(T_1, ..., T_4\). Boundary temperatures: top 20°, 20°; bottom 30°, 30°; left 10°, 10°; right 40°, 40°.
44 Systems of Linear Equations · Level 4
Solve the system of equations from Exercise 43. [Hint: To speed up the calculations, interchange rows 1 and 4 before starting "replace" operations.]

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