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Back to Pivots and their influence on solution spaces

Exercise 2

Avg difficulty 2.5

For each of the following linear systems, determine whether the system is consistent, and, if so, determine whether there are infinitely many solutions.

  1. 2x1−x2+3x3=10−x1+x2+3x4=82x2+2x3−x4=−43x1+2x2−x3+x4=10\begin{alignedat}{5} 2x_1&{}-{}& x_2& {}+{}& 3x_3 & {}{} & & {}={} & 10 \\ -x_1&{}+{}& x_2& & & {}+{} & 3x_4 & {}={} & 8 \\ &{}{}& 2x_2& {}+{}& 2x_3 & {}-{} & x_4 & {}={} & -4 \\ 3x_1&{}+{}& 2x_2& {}-{}& x_3 & {}+{} & x_4 & {}={} & 10 \\ \\ \end{alignedat}

  2. 2x1−x2+3x3+3x4=8−x1+x2+2x4=−12x2+2x3+6x4=43x1+2x2−x3−3x4=1\begin{alignedat}{5} 2x_1&{}-{}& x_2& {}+{}& 3x_3 & {}+{} & 3x_4 & {}={} & 8 \\ -x_1&{}+{}& x_2& & & {}+{} & 2x_4 & {}={} & -1 \\ &{}{}& 2x_2& {}+{}& 2x_3 & {}+{} & 6x_4 & {}={} & 4 \\ 3x_1&{}+{}& 2x_2& {}-{}& x_3 & {}-{} & 3x_4 & {}={} & 1 \\ \\ \end{alignedat}

  3. 2x1−x2+3x3+3x4=8−x1+x2+2x4=−12x2+2x3+6x4=43x1+2x2−x3−3x4=0\begin{alignedat}{5} 2x_1&{}-{}& x_2& {}+{}& 3x_3 & {}+{} & 3x_4 & {}={} & 8 \\ -x_1&{}+{}& x_2& & & {}+{} & 2x_4 & {}={} & -1 \\ &{}{}& 2x_2& {}+{}& 2x_3 & {}+{} & 6x_4 & {}={} & 4 \\ 3x_1&{}+{}& 2x_2& {}-{}& x_3 & {}-{} & 3x_4 & {}={} & 0 \\ \\ \end{alignedat}

Austin, Understanding Linear Algebra, §1.4 Pivots and their influence on solution spaces, Exercise 2

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Adapted from Understanding Linear Algebra © 2023 David Austin. Licensed under CC BY 4.0. Source: Understanding Linear Algebra. Imported at revision 3e6dced.