/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 Determine which of the following... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 7

Determine which of the following systems of linear equations has a solution. \(x_{1}+x_{2}-x_{3}+2 x_{4}=2\) (a) \(x_{1}+x_{2}+2 x_{3}=1\) (b) \(x_{1}+x_{2}-x_{3}=1\) \(2 x_{1}+2 x_{2}+x_{3}+2 x_{4}=4\) (b) \(x_{1}+x_{2}-x_{3}=1\) \(2 x_{1}+x_{2}+3 x_{3}=2\) \(x_{1}+2 x_{2}+3 x_{3}=1\) (d) \(x_{1}+x_{2}+3 x_{3}-x_{4}=0\) \(x_{1}+x_{2}+x_{3}+x_{4}=1\) $x_{1}-2 x_{2}+x_{3}-x_{4}=1$ (c) \(x_{1}+x_{2}-x_{3}=0\) \(x_{1}+2 x_{2}+x_{3}=3\) \(4 x_{1}+x_{2}+8 x_{3}-x_{4}=0\) (e) \(2 x_{1}+x_{2}+2 x_{3}=3\) \(x_{1}-4 x_{2}+7 x_{3}=4\)

Short Answer

Expert verified
System (a) has one solution, while systems (b), (c), (d), and (e) all have infinitely many solutions.

Step by step solution

01

System (a)

First, we'll examine system (a): 1. \(x_{1} + x_{2} - x_{3} + 2x_{4} = 2\) 2. \(x_{1} + x_{2} + 2x_{3} = 1\) Step 1: Put the two equations in matrix form: \[ \left[ \begin{array}{cccc|c} 1 & 1 & -1 & 2 & 2 \\ 1 & 1 & 2 & 0 & 1 \\ \end{array} \right] \] Step 2: Subtract the first row from the second row to eliminate \(x_{1}\) from the second equation: \[ \left[ \begin{array}{cccc|c} 1 & 1 & -1 & 2 & 2 \\ 0 & 0 & 3 & -2 & -1 \\ \end{array} \right] \] Since we now have a consistent system and we can solve for \(x_3\). So, system (a) has a solution.
02

System (b)

Now, we'll examine system (b): 1. \(x_{1} + x_{2} - x_{3} + 2x_{4} = 2\) 2. \(x_{1} + x_{2} - x_{3} = 1\) 3. \(2x_{1} + 2x_{2} + x_{3} + 2x_{4} = 4\) 4. \(x_{1} + x_{2} - x_{3} = 1\) We can see that equations 2 and 4 are equal, so we can eliminate the 4th equation, and we are left with a 3 equation - 4 variable system. Since we have fewer equations than variables, system (b) has infinitely many solutions.
03

System (c)

Now, we'll examine system (c): 1. \(x_{1} + x_{2} - x_{3} = 0\) 2. \(x_{1} + 2x_{2} + x_{3} = 3\) 3. \(4x_{1} + x_{2} + 8x_{3} - x_{4} = 0\) Since there are 4 variables and only 3 equations, system (c) has infinitely many solutions.
04

System (d)

Now, we'll examine system (d): 1. \(x_{1} + x_{2} + 3x_{3} - x_{4} = 0\) 2. \(x_{1} + x_{2} + x_{3} + x_{4} = 1\) 3. \(x_{1} - 2x_{2} + x_{3} - x_{4} = 1\) Since there are 4 variables and only 3 equations, system (d) has infinitely many solutions.
05

System (e)

Finally, let's examine system (e): 1. \(2x_{1} + x_{2} + 2x_{3} = 3\) 2. \(x_{1} - 4x_{2} + 7x_{3} = 4\) Since we have 3 variables and only 2 equations, system (e) has infinitely many solutions. #Conclusion#: System (a) has one solution, while systems (b), (c), (d), and (e) all have infinitely many solutions.

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Most popular questions from this chapter

Write \(v\) as a linear combination of \(u_{1}, u_{2}, u_{3},\) where (a) \(v=(3,10,7)\) and \(u_{1}=(1,3,-2), u_{2}=(1,4,2), u_{3}=(2,8,1)\) (b) \(v=(2,7,10)\) and \(u_{1}=(1,2,3), u_{2}=(1,3,5), u_{3}=(1,5,9)\) (c) \(v=(1,5,4)\) and \(u_{1}=(1,3,-2), u_{2}=(2,7,-1), u_{3}=(1,6,7)\)Find the equivalent system of linear equations by writing \(v=x u_{1}+y u_{2}+z u_{3}\). Alternatively, use the augmented matrix \(M\) of the equivalent system, where \(M=\left[u_{1}, u_{2}, u_{3}, v\right] .\) (Here \(u_{1}, u_{2}, u_{3}, v\) are the columns of \(M .)\)

Prove that if \(B\) is a \(3 \times 1\) matrix and \(C\) is a \(1 \times 3\) matrix, then the \(3 \times 3\) matrix \(B C\) has rank at most 1 . Conversely, show that if \(A\) is any \(3 \times 3\) matrix having rank 1 , then there exist a $3 \times 1\( matrix \)B\( and a \)1 \times 3\( matrix \)C\( such that \)A=B C$.

Suppose that the augmented matrix of a system \(A x=b\) is transformed into a matrix \(\left(A^{\prime} \mid b^{\prime}\right)\) in reduced row echelon form by a finite sequence of elementary row operations. (a) Prove that rank $\left(A^{\prime}\right) \neq \operatorname{rank}\left(A^{\prime} \mid b^{\prime}\right)$ if and only if \(\left(A^{\prime} \mid b^{\prime}\right)\) contains a row in which the only nonzero entry lies in the last column. (b) Deduce that \(A x=b\) is consistent if and only if $\left(A^{\prime} \mid b^{\prime}\right)$ contains no row in which the only nonzero entry lies in the last column.

Find the dimension and a basis of the general solution \(W\) of each of the following homogeneous systems: a. \(\begin{aligned} x-y+2 z &=0 \\ 2 x+y+z &=0 \\ 5 x+y+4 z &=0 \end{aligned}\) b. \(\begin{aligned} x+2 y-3 z &=0 \\ 2 x+5 y+2 z &=0 \\ 3 x-y-4 z &=0 \end{aligned}\) c. \(\begin{aligned} x+2 y+3 z+t &=0 \\ 2 x+4 y+7 z+4 t &=0 \\ 3 x+6 y+10 z+5 t &=0 \end{aligned}\)

Find the \(L U\) factorization of each of the following matrices: (a) \(\left[\begin{array}{ccc}1 & -1 & -1 \\ 3 & -4 & -2 \\ 2 & -3 & -2\end{array}\right]\) (c) \(\left[\begin{array}{lll}2 & 3 & 6 \\ 4 & 7 & 9 \\ 3 & 5 & 4\end{array}\right]\) , (d) \(\left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 4 & 7 \\ 3 & 7 & 10\end{array}\right]\) (b) \(\left[\begin{array}{rrr}1 & 3 & -1 \\ 2 & 5 & 1 \\ 3 & 4 & 2\end{array}\right]\)
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