11.6E: Solving Systems with Gaussian Elimination (Exercises)

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For the following exercises, write the system of linear equations from the augmented matrix. Indicate whether there will be a unique solution.

51. \(\left[\begin{array}{rrr|r}1 & 0 & -3 & 7 \\ 0 & 1 & 2 & -5 \\ 0 & 0 & 0 & 0\end{array}\right]\)

52. \(\left[\begin{array}{rrr|r}1 & 0 & 5 & -9 \\ 0 & 1 & -2 & 4 \\ 0 & 0 & 0 & 3\end{array}\right]\)

For the following exercises, write the augmented matrix from the system of linear equations.

53.

\(-2 x+2 y+z=7\)

\(2 x-8 y+5 z=0 \)

\(19 x-10 y+22 z=3\)

54.

\(4 x+2 y-3 z=14\)

\(-12 x+3 y+z=100\)

\(9 x-6 y+2 z=31\)

55.

\(x+3 z=12\)

\(-x+4 y=0\)

\(y+2 z=-7\)

For the following exercises, solve the system of linear equations using Gaussian elimination.

56.

\(3 x-4 y=-7\)
\(-6 x+8 y=14\)

\(3 x-4 y=1\)
\(-6 x+8 y=6\)

58.

\(-1.1 x-2.3 y=6.2\)
\(-5.2 x-4.1 y=4.3\)

59.

\(2 x+3 y+2 z=1\)

\(-4 x-6 y-4 z=-2\)
\(10 x+15 y+10 z=0\)

60.

\(-x+2 y-4 z=8\)

\(\quad 3 y+8 z=-4\)
\(-7 x+y+2 z=1\)