Explain elementary row operations and trace elimination in an augmented matrix.
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Try
Two tickets of one type plus one of another cost 11; one of the first plus one of the second costs 7. Write 2x+y=11 and x+y=7, where x,y are the two prices in the same currency. Subtraction gives x=4, and substitution gives y=3. The arithmetic works because each equation describes the same unknown quantities.
Predict
Predict whether subtracting the second equation from the first loses a solution. An augmented matrix records coefficients and the right-hand side together: rows [2,1|11] and [1,1|7]. The bar separates the coefficient matrix A from vector b in Ax=b. Rows represent equations; columns represent variables in a fixed order.
Experiment
Swap the rows, then replace row 2 by row 2 minus twice row 1. Write every entry, including the right-hand side. The rows become [1,1|7] and [0,−1|−3]. Back-substitute to recover y=3 and x=4. Check the original equations, not only the reduced ones.
Equal coordinate scales: x+y=7 and 2x+y=11 meet at (4,3). The intersection checks the price solution; elimination supplies the exact argument.
Understand
Worked general reasoning: swapping equations changes only their order. Multiplying an equation by a nonzero scalar is reversible by dividing by that scalar. Replacing E₂ by E₂+kE₁ while keeping E₁ is reversible by subtracting kE₁. These three elementary operations therefore preserve the entire solution set.
Elimination chooses a nonzero pivot, clears entries below it and continues on the remaining rows and columns. Echelon form has successive pivots farther right and zero rows at the bottom. Back-substitution then starts with the last nonzero row. Reduced row-echelon form additionally has unit pivots and zeros above them.
Invalid step: multiplying a row by zero discards a constraint. For example x=4 becomes 0=0, admitting every x. A visually simpler matrix is useful only if the transformation remains reversible.
Apply
Annotate each practice row operation and verify dimensions. Fractions are legitimate exact coefficients; replacing them by rounded decimals can spoil a precise consistency check. The symbolic argument about reversibility applies to any finite system, not only two equations.
Continue
Depth: eliminating several right-hand sides alongside one coefficient matrix reuses the same operations. Later inverse and least-squares computations exploit that structure. Numerical software also chooses pivots to limit roundoff amplification; exact algebraic validity and numerical stability are different questions.
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Solve x+y=7 and 2x−y=5. Enter x.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Replace row [3,5|11] by that row minus 3 times [1,2|4]. Enter its new right-hand entry.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Which operation always preserves a system's solution set?
When subtracting one equation from another, what happens to the right-hand side?
Solve x+y+z=6, y+z=4, z=1, and check all original equations.
Structure your argument and preserve revisions
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Prove replacing equation E2 by E2+kE1 preserves solutions when E1 is kept.
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.
