The system of linear equations [math: 2x1+3x2+x3+x5=−1−4x1−6x2−2x3−4x4+18x5=2610x1+15x2+5x3−x4+10x5=1] has row echelon form [math: [23101−1000−156000000].] How many solutions does this system have?

Question

BlackTom AI · Accepted Answer

Pivot columns are x1 and x4, so the coefficient matrix has rank 2.

1) Relevant concepts: rank (number of nonzero rows/pivots in row-echelon form), number of variables n, free variables = n − rank, consistency (no contradictory row [0 ... 0 | b] with b ≠ 0). If the system is consistent and has at least one free variable, it has infinitely many solutions.

2) Reasoning step-by-step:
- The given row-echelon augmented matrix is
  [ 2  3  1  0  1 | −1 ]
  [ 0  0  0 −1  5 |  6 ]
  [ 0  0  0  0  0 |  0 ].
- There are two nonzero rows (first and second), so rank = 2. Number of unknowns n = 5 (x1,...,x5).
- Check consistency: the third row is 0 = 0, so no contradiction; the system is consistent.

3) Calculations / intermediate steps:
- Free variables = n − rank = 5 − 2 = 3. Choose free variables x2, x3, x5.
- Solve pivot equations for pivot variables:
  From row1: 2x1 + 3x2 + x3 + x5 = −1 → x1 = (−1 − 3x2 − x3 − x5)/2.
  From row2: −x4 + 5x5 = 6 → x4 = 5x5 − 6.
- Parameterize: let x2 = s, x3 = t, x5 = u (arbitrary real parameters). Then x1 and x4 given above. This yields infinitely many solutions (one for each triple (s,t,u)).

4) Why the correct answer is correct:
- The system is consistent and has 3 free parameters, so there are infinitely many solutions (a 3-parameter family).

5) Why other choices are incorrect (brief):
- “five” or “three”: these count numbers of solutions incorrectly; finite counts >1 don’t arise here because free parameters produce infinitely many, not a finite fixed count.
- “two”, “one”: these imply a finite number of solutions; not possible since free variables > 0 and system consistent.
- “none”: would require an inconsistent row like [0 ... 0 | b≠0], which is not present.

6) Conclusion: therefore the system has infinitely many solutions. This confirms answer d.

The system of linear equations [math: 2x1+3x2+x3+x5=−1−4x1−6x2−2x3−4x4+18x5=2610x1+15x2+5x3−x4+10x5=1] has row echelon form [math: [23101−1000−156000000].] How many solutions does this system have?单项选择题

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