Collection of Maths Problems

Systems Ib.

Task number: 2383

• Variant 1

  \( \begin{array}{rrrrrr} 17 a& +9 b& -9 c& +3d&=& -8 \\ 11 a& & +2 c& &=& -7 \\ 13 a& +2 b& - c& &=& -9 \\ 7 a& +3 b& -5 c& + d&=& -8 \end{array} \)

• Hint

  Use Gaussian elimination, but use another order of columns, otherwise you get bad numbers.

• Resolution

  \[\begin{multline*} \left( \begin{array}{cccc|c} 17 & 9 & -9 & 3 & -8 \\ 11 & 0 & 2 & 0 & -7 \\ 13 & 2 & -1 & 0 & -9 \\ 7 & 3 & -5 & 1 & -8 \end{array} \right) \sim \left( \begin{array}{cccc|c} -4 & 0 & 6 & 0 & 16 \\ 11 & 0 & 2 & 0 & -7 \\ 13 & 2 & -1 & 0 & -9 \\ 7 & 3 & -5 & 1 & -8 \end{array} \right) \sim \left( \begin{array}{cccc|c} -37& 0 & 0 & 0 & 37 \\ 11 & 0 & 2 & 0 & -7 \\ 13 & 2 & -1 & 0 & -9 \\ -25& 0 & -7 & 2 & 11 \end{array} \right) \sim \\ \left( \begin{array}{cccc|c} 1 & 0 & 0 & 0 & -1 \\ 0 & 0 & 2 & 0 & 4 \\ 0 & 2 & -1 & 0 & 4 \\ 0 & 0 & -7 & 2 & 14 \end{array} \right) \sim \left( \begin{array}{cccc|c} 1 & 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 & 2 \\ 0 & 2 & 0 & 0 & 6 \\ 0 & 0 & 0 & 2 & 0 \end{array} \right) \sim \left( \begin{array}{cccc|c} 1 & 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 & 2 \\ 0 & 1 & 0 & 0 & 3 \\ 0 & 0 & 0 & 1 & 0 \end{array} \right) \end{multline*}\]

• Result

  The solution of the system is \(a = -1\), \(b = 3\), \(c = 2\), \(d = 0\).

• Variant 2

  \( \begin{array}{rrrrrr} 17 a& +8 b& +4 c& +7d&=& 30 \\ -11 a& & & +3d &=& -2 \\ 7 a& +2 b& + c& -2d &=& -1 \\ 13 a& +6 b& & -2d&=& 7 \end{array} \)

• Resolution

  \[\begin{multline*} \left( \begin{array}{cccc|c} 17 & 8 & 4 & 7 & 30 \\ -11 & 0 & 0 & 3 & -2 \\ 7 & 2 & 1 & -2 & -1 \\ 13 & 6 & 0 & -2 & 7 \end{array} \right) \sim \left( \begin{array}{cccc|c} -11 & 0 & 0 & 15 & 34 \\ -11 & 0 & 0 & 3 & -2 \\ 7 & 2 & 1 & -2 & -1 \\ 13 & 6 & 0 & -2 & 7 \end{array} \right) \sim \left( \begin{array}{cccc|c} 0 & 0 & 0 & 12 & 36 \\ -11 & 0 & 0 & 3 & -2 \\ 8 & 0 & 3 & -4 & -10 \\ 13 & 6 & 0 & -2 & 7 \end{array} \right) \sim \\ \left( \begin{array}{cccc|c} 0 & 0 & 0 & 1 & 3 \\ -11 & 0 & 0 & 0 & -11 \\ 8 & 0 & 3 & 0 & 2 \\ 13 & 6 & 0 & 0 & 13 \end{array} \right) \sim \left( \begin{array}{cccc|c} 0 & 0 & 0 & 1 & 3 \\ 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 3 & 0 & -6 \\ 0 & 6 & 0 & 0 & 0 \end{array} \right) \sim \left( \begin{array}{cccc|c} 0 & 0 & 0 & 1 & 3 \\ 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & -2 \\ 0 & 1 & 0 & 0 & 0 \end{array} \right) \end{multline*}\]

• Result

  The solution of the system is \(a = 1\), \(b = 0\), \(c = -2\), \(d = 3\).