1.4.1: Ejercicios 1.4
- Page ID
- 116484
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)En Ejercicios\(\PageIndex{1}\) -\(\PageIndex{14}\), encuentra la solución al sistema lineal dado. Si el sistema tiene una solución infinita, dé 2 soluciones particulares.
\(\begin{array}{ccccc} 2x_1&+&4x_2&=&2\\ x_1&+&2x_2&=&1\\ \end{array}\)
- Contestar
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\(x_1=1-2x_2\); \(x_2\) is free. Possible solutions: \(x_1=1\), \(x_2=0\) and \(x_1=-1\), \(x_2=1\).
\(\begin{array}{ccccc} -x_1&+&5x_2&=&3\\ 2x_1&-&10x_2&=&-6\\ \end{array}\)
- Contestar
-
\(x_1=-3+5x_2\); \(x_2\) is free. Possible solutions: \(x_1 = 3\), \(x_2=0\) and \(x_1 = -8\), \(x_2 = -1\)
\(\begin{array}{ccccc} x_1&+&x_2&=&3\\ 2x_1&+&x_2&=&4\\ \end{array}\)
- Contestar
-
\(x_1=1\); \(x_2=2\)
\(\begin{array}{ccccc} -3x_1&+&7x_2&=&-7\\ 2x_1&-&8x_2&=&8\\ \end{array}\)
- Contestar
-
\(x_1=0\); \(x_2=-1\)
\(\begin{array}{ccccc} 2x_1&+&3x_2&=&1\\ -2x_1&-&3x_2&=&1\\ \end{array}\)
- Contestar
-
No hay solución; el sistema es inconsistente.
\(\begin{array}{ccccc} x_1&+&2x_2&=&1\\ -x_1&-&2x_2&=&5\\ \end{array}\)
- Contestar
-
No hay solución; el sistema es inconsistente.
\(\begin{array}{ccccccc} -2x_1&+&4x_2&+&4x_3&=&6\\ x_1&-&3x_2&+&2x_3&=&1\\ \end{array}\)
- Contestar
-
\(x_1=-11+10x_3\); \(x_2=-4+4x_3\); \(x_3\) is free. Possible solutions: \(x_1=-11\), \(x_2 = -4\), \(x_3=0\) and \(x_1 = -1\), \(x_2 = 0\) and \(x_3 = 1\).
\(\begin{array}{ccccccc} -x_1&+&2x_2&+&2x_3&=&2\\ 2x_1&+&5x_2&+&x_3&=&2\\ \end{array}\)
- Contestar
-
\(x_1=-\frac23+\frac89x_3\); \(x_2=\frac23-\frac59x_3\); \(x_3\) is free. Possible solutions: \(x_1 = -\frac23\), \(x_2 = \frac23\), \(x_3 = 0\) and \(x_1 = \frac49\), \(x_2 = -\frac19\), \(x_3 = 1\)
\(\begin{array}{rcl} -x_1-x_2+x_3+x_4&=&0\\ -2x_1-2x_2+x_3&=&-1\\ \end{array}\)
- Contestar
-
\(x_1=1-x_2-x_4\); \(x_2\) is free; \(x_3=1-2x_4\); \(x_4\) is free. Possible solutions: \(x_1 = 1\), \(x_2 = 0\), \(x_3 = 1\), \(x_4 = 0\) and \(x_1 = -2\), \(x_2 = 1\), \(x_3 = -3\), \(x_4=2\)
\(\begin{array}{rcl} x_1+x_2+6x_3+9x_4&=&0\\ -x_1-x_3-2x_4&=&-3\\ \end{array}\)
- Contestar
-
\(x_1=3-x_3-2x_4\); \(x_2=-3-5x_3-7x_4\); \(x_3\) is free; \(x_4\) is free. Possible solutions: \(x_1 =3\), \(x_2 = -3\), \(x_3=0\), \(x_4=0\) and \(x_1 = 0\), \(x_2 = -5\), \(x_3 =-1\), \(x_4=1\)
\(\begin{array}{ccccccc} 2x_1&+&x_2&+&2x_3&=&0\\ x_1&+&x_2&+&3x_3&=&1\\ 3x_1&+&2x_2&+&5x_3&=&3\\ \end{array}\)
- Contestar
-
No hay solución; el sistema es inconsistente.
\(\begin{array}{ccccccc} x_1&+&3x_2&+&3x_3&=&1\\ 2x_1&-&x_2&+&2x_3&=&-1\\ 4x_1&+&5x_2&+&8x_3&=&2\\ \end{array}\)
- Contestar
-
No hay solución; el sistema es inconsistente.
\(\begin{array}{ccccccc} x_1&+&2x_2&+&2x_3&=&1\\ 2x_1&+&x_2&+&3x_3&=&1\\ 3x_1&+&3x_2&+&5x_3&=&2\\ \end{array}\)
- Contestar
-
\(x_1=\frac13-\frac43x_3\); \(x_2=\frac13-\frac13x_3\); \(x_3\) is free. Possible solutions: \(x_1 = \frac13\), \(x_2=\frac13\), \(x_3=0\) and \(x_1 = -1\), \(x_2 = 0\), \(x_3=1\)
\(\begin{array}{ccccccc} 2x_1&+&4x_2&+&6x_3&=&2\\ 1x_1&+&2x_2&+&3x_3&=&1\\ -3x_1&-&6x_2&-&9x_3&=&-3\\ \end{array}\)
- Contestar
-
\(x_1=1-2x_2-3x_3\); \(x_2\) is free; \(x_3\) is free. Possible solutions: \(x_1=1\), \(x_2=0\), \(x_3=0\) and \(x_1=8\), \(x_2=1\), \(x_3 = -3\)
En Ejercicios\(\PageIndex{15}\) -\(\PageIndex{18}\), estado para qué valores\(k\) del sistema dado tendrán exactamente 1 solución, infinitas soluciones, o ninguna solución.
\(\begin{array}{ccccc}x_1&+&2x_2&=&1\\2x_1&+&4x_2&=&k\end{array}\)
- Contestar
-
Nunca exactamente 1 solución; infinitas soluciones si\(k=2\); no solution if \(k\neq 2\).
\(\begin{array}{ccccc}x_1&+&2x_2&=&1\\x_1&+&kx_2&=&1\end{array}\)
- Contestar
-
Exactamente 1 solución si\(k\neq 2\); infinite solutions if \(k=2\); never no solution.
\(\begin{array}{ccccc}x_1&+&2x_2&=&1\\x_1&+&kx_2&=&2\end{array}\)
- Contestar
-
Exactamente 1 solución si\(k\neq 2\); no solution if \(k=2\); never infinite solutions.
\(\begin{array}{ccccc}x_1&+&2x_2&=&1\\x_1&+&3x_2&=&k\end{array}\)
- Contestar
-
Exactamente 1 solución para todos\(k\).