Systems and Simultaneous Constraints: cumulative review

Repair a solution that skips “Solve $4x + 2(-x + 5) = 6$ to obtain $x = -2$.” while solving: Solve $4x + 2y = 6$ and $y = -x + 5$ by substitution.

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In this substitution case, change one numerical value, solve the revised problem, and identify which parts of the original method still apply: Solve $y = 3x - 4$ and $2x + y = 11$ by substitution.

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Connect the opening situation “Use one pricing rule inside another total equation.” to the algebraic structure used in the worked case. Define quantities and units before writing any equation.

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Explain why the method for substitution is valid here and name one nearby problem where it would not apply.

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Compare the conclusions of all three worked cases with this lesson outcome—Replace one variable expression with an equal expression to reduce a system. Explain what remains invariant across them.

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Exit check: solve and verify without referring to the displayed steps. Solve $x = 2y + 1$ and $3x - y = 13$ by substitution.

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Exit check: solve and verify without referring to the displayed steps. Solve $4x + 2y = 6$ and $y = -x + 5$ by substitution.

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Classify the mathematical object and requested action in this lesson case: Solve $2x + 3y = 13$ and $4x - 3y = 5$ by elimination.

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State the central definition behind this outcome: Combine scaled equations to remove a variable while preserving the common solution.

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Before calculating, list every sign, endpoint, unit, grouping, or domain condition that can affect: Solve $2x + 3y = 13$ and $4x - 3y = 5$ by elimination.

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Explain why this opening move is valid: Add the equations so the opposite y-terms cancel.

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Solve $2x + 3y = 13$ and $4x - 3y = 5$ by elimination.

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Solve $3x + 2y = 16$ and $5x - 2y = 8$ by elimination.

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Solve $2x + 3y = 7$ and $5x + 2y = 8$ by elimination.

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Verify the proposed result “$(3, \frac{7}{3})$.” against the original statement.

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Complete the calculation after “Add the equations to eliminate $y$.” in this problem: Solve $3x + 2y = 16$ and $5x - 2y = 8$ by elimination.

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Name and justify the most efficient first move, then solve: Solve $2x + 3y = 7$ and $5x + 2y = 8$ by elimination.

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Compare the methods used in these two cases and identify the structural reason they differ: Solve $3x + 2y = 16$ and $5x - 2y = 8$ by elimination. Solve $2x + 3y = 7$ and $5x + 2y = 8$ by elimination.

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Create the representation most useful for checking this result: Solve $3x + 2y = 16$ and $5x - 2y = 8$ by elimination. Connect the pair of equations, their graph or feasible regions, and the ordered-pair check.

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A learner reports “$(3, \frac{7}{3})$.” but omits the original-condition check. Explain the risk before deciding whether the result is supported.

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Repair a solution that skips “Add $4x + 6y = 14$ and $-15x - 6y = -24$ to obtain $-11x = -10$.” while solving: Solve $2x + 3y = 7$ and $5x + 2y = 8$ by elimination.

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In this elimination case, change one numerical value, solve the revised problem, and identify which parts of the original method still apply: Solve $2x + 3y = 13$ and $4x - 3y = 5$ by elimination.

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Connect the opening situation “Cancel one unknown by combining two measurements.” to the algebraic structure used in the worked case. Define quantities and units before writing any equation.

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Explain why the method for elimination is valid here and name one nearby problem where it would not apply.

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Compare the conclusions of all three worked cases with this lesson outcome—Combine scaled equations to remove a variable while preserving the common solution. Explain what remains invariant across them.

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