1.4.1: Exercises 1.4

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In Exercises \(\PageIndex{1}\) - \(\PageIndex{14}\), find the solution to the given linear system. If the system has infinite solution, give 2 particular solutions.

Exercise \(\PageIndex{1}\)

\(\begin{array}{ccccc} 2x_1&+&4x_2&=&2\\ x_1&+&2x_2&=&1\\ \end{array}\)

Answer: \(x_1=1-2x_2\); \(x_2\) is free. Possible solutions: \(x_1=1\), \(x_2=0\) and \(x_1=-1\), \(x_2=1\).

Exercise \(\PageIndex{2}\)

\(\begin{array}{ccccc} -x_1&+&5x_2&=&3\\ 2x_1&-&10x_2&=&-6\\ \end{array}\)

Answer: \(x_1=-3+5x_2\); \(x_2\) is free. Possible solutions: \(x_1 = 3\), \(x_2=0\) and \(x_1 = -8\), \(x_2 = -1\)

Exercise \(\PageIndex{3}\)

\(\begin{array}{ccccc} x_1&+&x_2&=&3\\ 2x_1&+&x_2&=&4\\ \end{array}\)

Answer: \(x_1=1\); \(x_2=2\)

Exercise \(\PageIndex{4}\)

\(\begin{array}{ccccc} -3x_1&+&7x_2&=&-7\\ 2x_1&-&8x_2&=&8\\ \end{array}\)

Answer: \(x_1=0\); \(x_2=-1\)

Exercise \(\PageIndex{5}\)

\(\begin{array}{ccccc} 2x_1&+&3x_2&=&1\\ -2x_1&-&3x_2&=&1\\ \end{array}\)

Answer: No solution; the system is inconsistent.

Exercise \(\PageIndex{6}\)

\(\begin{array}{ccccc} x_1&+&2x_2&=&1\\ -x_1&-&2x_2&=&5\\ \end{array}\)

Answer: No solution; the system is inconsistent.

Exercise \(\PageIndex{7}\)

\(\begin{array}{ccccccc} -2x_1&+&4x_2&+&4x_3&=&6\\ x_1&-&3x_2&+&2x_3&=&1\\ \end{array}\)

Answer: \(x_1=-11+10x_3\); \(x_2=-4+4x_3\); \(x_3\) is free. Possible solutions: \(x_1=-11\), \(x_2 = -4\), \(x_3=0\) and \(x_1 = -1\), \(x_2 = 0\) and \(x_3 = 1\).

Exercise \(\PageIndex{8}\)

\(\begin{array}{ccccccc} -x_1&+&2x_2&+&2x_3&=&2\\ 2x_1&+&5x_2&+&x_3&=&2\\ \end{array}\)

Answer: \(x_1=-\frac23+\frac89x_3\); \(x_2=\frac23-\frac59x_3\); \(x_3\) is free. Possible solutions: \(x_1 = -\frac23\), \(x_2 = \frac23\), \(x_3 = 0\) and \(x_1 = \frac49\), \(x_2 = -\frac19\), \(x_3 = 1\)

Exercise \(\PageIndex{9}\)

\(\begin{array}{rcl} -x_1-x_2+x_3+x_4&=&0\\ -2x_1-2x_2+x_3&=&-1\\ \end{array}\)

Answer: \(x_1=1-x_2-x_4\); \(x_2\) 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 \(x_1 = -2\), \(x_2 = 1\), \(x_3 = -3\), \(x_4=2\)

Exercise \(\PageIndex{10}\)

\(\begin{array}{rcl} x_1+x_2+6x_3+9x_4&=&0\\ -x_1-x_3-2x_4&=&-3\\ \end{array}\)

Answer: \(x_1=3-x_3-2x_4\); \(x_2=-3-5x_3-7x_4\); \(x_3\) is free; \(x_4\) is free. Possible solutions: \(x_1 =3\), \(x_2 = -3\), \(x_3=0\), \(x_4=0\) and \(x_1 = 0\), \(x_2 = -5\), \(x_3 =-1\), \(x_4=1\)

Exercise \(\PageIndex{11}\)

\(\begin{array}{ccccccc} 2x_1&+&x_2&+&2x_3&=&0\\ x_1&+&x_2&+&3x_3&=&1\\ 3x_1&+&2x_2&+&5x_3&=&3\\ \end{array}\)

Answer: No solution; the system is inconsistent.

Exercise \(\PageIndex{12}\)

\(\begin{array}{ccccccc} x_1&+&3x_2&+&3x_3&=&1\\ 2x_1&-&x_2&+&2x_3&=&-1\\ 4x_1&+&5x_2&+&8x_3&=&2\\ \end{array}\)

Answer: No solution; the system is inconsistent.

Exercise \(\PageIndex{13}\)

\(\begin{array}{ccccccc} x_1&+&2x_2&+&2x_3&=&1\\ 2x_1&+&x_2&+&3x_3&=&1\\ 3x_1&+&3x_2&+&5x_3&=&2\\ \end{array}\)

Answer: \(x_1=\frac13-\frac43x_3\); \(x_2=\frac13-\frac13x_3\); \(x_3\) is free. Possible solutions: \(x_1 = \frac13\), \(x_2=\frac13\), \(x_3=0\) and \(x_1 = -1\), \(x_2 = 0\), \(x_3=1\)

Exercise \(\PageIndex{14}\)

\(\begin{array}{ccccccc} 2x_1&+&4x_2&+&6x_3&=&2\\ 1x_1&+&2x_2&+&3x_3&=&1\\ -3x_1&-&6x_2&-&9x_3&=&-3\\ \end{array}\)

Answer: \(x_1=1-2x_2-3x_3\); \(x_2\) is free; \(x_3\) is free. Possible solutions: \(x_1=1\), \(x_2=0\), \(x_3=0\) and \(x_1=8\), \(x_2=1\), \(x_3 = -3\)

In Exercises \(\PageIndex{15}\) - \(\PageIndex{18}\), state for which values of \(k\) the given system will have exactly 1 solution, infinite solutions, or no solution.

Exercise \(\PageIndex{15}\)

\(\begin{array}{ccccc}x_1&+&2x_2&=&1\\2x_1&+&4x_2&=&k\end{array}\)

Answer: Never exactly 1 solution; infinite solutions if \(k=2\); no solution if \(k\neq 2\).

Exercise \(\PageIndex{16}\)

\(\begin{array}{ccccc}x_1&+&2x_2&=&1\\x_1&+&kx_2&=&1\end{array}\)

Answer: Exactly 1 solution if \(k\neq 2\); infinite solutions if \(k=2\); never no solution.

Exercise \(\PageIndex{17}\)

\(\begin{array}{ccccc}x_1&+&2x_2&=&1\\x_1&+&kx_2&=&2\end{array}\)

Answer: Exactly 1 solution if \(k\neq 2\); no solution if \(k=2\); never infinite solutions.

Exercise \(\PageIndex{18}\)

\(\begin{array}{ccccc}x_1&+&2x_2&=&1\\x_1&+&3x_2&=&k\end{array}\)

Answer: Exactly 1 solution for all \(k\).