Distinguish no solution, one solution and a family
Read pivots, contradictions and free variables without counting equations alone.
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Try
The equations x+y=3 and 2x+2y=6 describe the same line, so there are infinitely many solutions. Change only the second right-hand side to 7: subtraction produces 0=1 and no solution remains. The coefficient rows look equally dependent in both cases; consistency also depends on the right-hand side.
Predict
Predict whether having as many equations as unknowns guarantees uniqueness. It does not. A pivot variable leads a nonzero row; a free variable belongs to a nonpivot coefficient column. First check for a contradiction, then count free variables. A pivot in the augmented column represents inconsistency, not an extra unknown.
Experiment
Reduce rows [1,2,−1|3] and [2,4,−2|6]. Name the free variables y=s and z=t and generate two different solutions. Then change the final 6 to 7 and repeat. Explain why parameterizing the first row cannot repair the contradictory second row.
Understand
Worked family: x−y+2z=4, y+z=3 gives z=t, y=3−t and x=7−3t. Thus every solution is (7,3,0)+t(−3,−1,1), for real t. Substitution verifies each generated point, and solving backward shows that no solution is omitted.
For a consistent Ax=b, every free variable can vary independently and each pivot variable is determined by them. With n columns and r pivots, there are n−r parameters. No free variables means a unique solution. A homogeneous system Ax=0 is always consistent because zero works.
Invalid inference: a zero row always means infinitely many solutions. A three-equation system in two unknowns can have two pivots plus one redundant zero row and still have one solution.
Apply
State the classification before giving a formula. For a family, name the real parameters and show both that the formula satisfies the equations and that any solution has that form. A finite list of sample points is not the whole solution set.
Continue
Depth: a nonempty solution set is a translate of the homogeneous kernel. It is a subspace only if it contains zero, which requires b=0. This distinction will recur when comparing affine geometric constraints with linear subspaces.
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Practice and reasoning
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A consistent echelon system has 5 unknowns and 3 pivot columns. How many free variables?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A system reduces to x+2z=5 and y−z=1. For z=3, enter x.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Which augmented row certifies inconsistency?
A zero row in a consistent system means:
Describe all solutions of x+2y−z=3 and 2x+4y−2z=6 using two real parameters.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
If Ax=b has one solution p, prove its full solution set is p+ker A.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
