Testing (3, 2) in y = -2x + 8
A point solves y = -2x + 8 when substituting its x-coordinate yields exactly its y-coordinate; (3, 2) does not, because -2(3)+8 = 2 only by coincidence of arithmetic but equals 2, which equals the y-value, so it actually is a solution.
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Is (3, 2) a solution to y = -2x + 8?
Someone claims (3, 2) is a solution to y = -2x + 8 — but plugging in gives something that looks close. Is it really a solution?
Students often substitute x and y separately and feel satisfied when the numbers 'look right,' without checking the equation as a whole.
Step-by-step substitution showing the right-hand side result vs. the given y value, with a side-by-side comparison.
A point (x, y) solves y = -2x + 8 only if substituting x produces exactly the given y; (3, 2) fails this check.
- graphing the line
- solving systems of two equations
- slope-intercept form derivation
- 01Does (3, 2) Fit the Line?slideSlot 1Hook
Present the equation y = -2x + 8 and the candidate point (3, 2). Ask whether this point satisfies the equation.
- Equation: y = -2x + 8
- Candidate point: (3, 2)
- Question: does this point belong on the line?
PhenomenonA point (3, 2) is offered as a possible solution to y = -2x + 8.
QuestionIf we drop (3, 2) onto the line, does it land exactly on the line?
- 02The Quick GuessslideSlot 2Tension
Surface the common shortcut: checking x and y separately without combining them through the equation.
- x = 3 and y = 2 are both small numbers
- Numbers feel 'plausible' for this line
- But plausibility is not proof — substitution is required
PredictionMost learners will predict 'yes' based on the small numbers, expecting the equation to behave simply.
Tempting intuitionIf x and y both look reasonable, the point must satisfy the equation.
- 03Substitute and CompareslideSlot 3Reveal
Walk through substituting x = 3 into -2x + 8, then compare the result with y = 2 to decide.
- Step 1: Replace x with 3: y = -2(3) + 8 = -6 + 8 = 2
- Step 2: The right-hand side gives y = 2
- Step 3: The point's y-coordinate is also 2
- Conclusion: the two values match, so (3, 2) is a solution
EvidenceComputing -2(3) + 8 yields 2, which is exactly the y-coordinate of the point.
ConclusionBecause the computed y equals the point's y, (3, 2) is indeed a solution to y = -2x + 8.
Mechanism- 1Substitute x = 3 into the expression -2x + 8, which forces the right-hand side to evaluate to a single number.
- 2Compare that computed number with the given y; equality means the point lies on the line.
- 04The Substitution RuleslideSlot 4Takeaway
Apply the same substitution test to a new point so the learner transfers the verification habit.
- Rule: substitute x, compute the right-hand side, compare with y
- New check: is (2, 4) a solution to y = -2x + 8?
- Compute -2(2) + 8 = 4, which matches y = 4, so yes
TransferUse the same substitution procedure on a different point such as (2, 4) in y = -2x + 8.
Expected inferenceThe learner computes -2(2)+8 = 4 and confirms it equals the y-coordinate, so (2, 4) is also a solution.
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