Systems and Simultaneous Constraints: mastery check

Complete the calculation after “Substitute into the first equation: $3(2) + (-1) = 5$.” in this problem: Determine whether $(2, -1)$ solves $3x + y = 5$ and $x - 2y = 4$.

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Name and justify the most efficient first move, then solve: Find the value of $k$ for which $(3, 2)$ solves $2x +$ ky $= 14$ and verify it.

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Compare the methods used in these two cases and identify the structural reason they differ: Determine whether $(2, -1)$ solves $3x + y = 5$ and $x - 2y = 4$. Find the value of $k$ for which $(3, 2)$ solves $2x +$ ky $= 14$ and verify it.

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Create the representation most useful for checking this result: Determine whether $(2, -1)$ solves $3x + y = 5$ and $x - 2y = 4$. Connect the pair of equations, their graph or feasible regions, and the ordered-pair check.

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A learner reports “Yes; $4 + 3 = 7$ and $2(4) - 3 = 5$.” 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 “Solve $6 + 2k = 14$ to obtain $k = 4$.” while solving: Find the value of $k$ for which $(3, 2)$ solves $2x +$ ky $= 14$ and verify it.

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In this systems as simultaneous conditions case, change one numerical value, solve the revised problem, and identify which parts of the original method still apply: Determine whether $(4, 3)$ solves $x + y = 7$ and $2x - y = 5$.

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Connect the opening situation “Test a candidate plan against two requirements.” 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 systems as simultaneous conditions 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—Understand a solution as an ordered pair satisfying every equation or constraint. Explain what remains invariant across them.

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Exit check: solve and verify without referring to the displayed steps. Determine whether $(2, -1)$ solves $3x + y = 5$ and $x - 2y = 4$.

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Exit check: solve and verify without referring to the displayed steps. Find the value of $k$ for which $(3, 2)$ solves $2x +$ ky $= 14$ and verify it.

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Classify the mathematical object and requested action in this lesson case: Use the graphs $y = 2x + 1$ and $y = -x + 7$ to identify their intersection, then verify algebraically.

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State the central definition behind this outcome: Read intersections and estimate exact or approximate system solutions.

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Before calculating, list every sign, endpoint, unit, grouping, or domain condition that can affect: Use the graphs $y = 2x + 1$ and $y = -x + 7$ to identify their intersection, then verify algebraically.

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Explain why this opening move is valid: Set the two output expressions equal because an intersection has the same y-value.

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Use the graphs $y = 2x + 1$ and $y = -x + 7$ to identify their intersection, then verify algebraically.

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Find the intersection of$$y = -2x + 9 \qquad y = x - 3$$

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