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In a certain learning experiment, each participant had three trials and was assigned, for each trial, a score of either - 2, - 1, 0, 1, or 2 . The participant's final score consisted of the sum of the first trial score, 2 times the second trial score, and 3 times the third trial score. If Anne received scores of 1 and -1 for her first two trials, not necessarily in that order, which of the following could NOT be her final score?

Explanation	Calculations	Help
$We first name the trial scores x, y, and z, and write the final score S as a literal expression so we can test each case.$	$S = x + 2 \times y + 3 \times z x is either 1 or - 1 y is either 1 or - 1 z is one of - 2, - 1, 0, 1, or 2$	Theory & Tactics Method Card CALC-3A Click to view full details
$We then try the case where the first trial gives a 1 and the second gives a -1 to simplify S in terms of z.$	$S = 1 + 2 \times (- 1) + 3 \times z S = - 1 + 3 \times z$
$We then list each possible third score and find the resulting S in this first case.$	$z = - 2, gives S = - 1 + 3 \times (- 2) = - 7 z = - 1, gives S = - 1 + 3 \times (- 1) = - 4 z = 0, gives S = - 1 + 3 \times 0 = - 1 z = 1, gives S = - 1 + 3 \times 1 = 2 z = 2, gives S = - 1 + 3 \times 2 = 5$
$We then try the case where the first trial gives a -1 and the second gives a 1 to get a new expression for S.$	$S = - 1 + 2 \times 1 + 3 \times z S = 1 + 3 \times z$
$We then list each possible third score and find S in this second case.$	$z = - 2, gives S = 1 + 3 \times (- 2) = - 5 z = - 1, gives S = 1 + 3 \times (- 1) = - 2 z = 0, gives S = 1 + 3 \times 0 = 1 z = 1, gives S = 1 + 3 \times 1 = 4 z = 2, gives S = 1 + 3 \times 2 = 7$
$We combine the scores from both cases and see which answer choice does not appear.$	$- 7, - 5, - 4, - 2, - 1, 1, 2, 4, 5, and 7$

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In a certain learning experiment, each participant had three trials and was assigned, for each trial, a score of either - 2, - 1, 0, 1, or 2 . The participant's final score consisted of the sum of the first trial score, 2 times the second trial score, and 3 times the third trial score. If Anne received scores of 1 and -1 for her first two trials, not necessarily in that order, which of the following could NOT be her final score?

Explanation	Calculations	Help
$We first name the trial scores x, y, and z, and write the final score S as a literal expression so we can test each case.$	$S = x + 2 \times y + 3 \times z x is either 1 or - 1 y is either 1 or - 1 z is one of - 2, - 1, 0, 1, or 2$	Theory & Tactics Method Card CALC-3A Click to view full details
$We then try the case where the first trial gives a 1 and the second gives a -1 to simplify S in terms of z.$	$S = 1 + 2 \times (- 1) + 3 \times z S = - 1 + 3 \times z$
$We then list each possible third score and find the resulting S in this first case.$	$z = - 2, gives S = - 1 + 3 \times (- 2) = - 7 z = - 1, gives S = - 1 + 3 \times (- 1) = - 4 z = 0, gives S = - 1 + 3 \times 0 = - 1 z = 1, gives S = - 1 + 3 \times 1 = 2 z = 2, gives S = - 1 + 3 \times 2 = 5$
$We then try the case where the first trial gives a -1 and the second gives a 1 to get a new expression for S.$	$S = - 1 + 2 \times 1 + 3 \times z S = 1 + 3 \times z$
$We then list each possible third score and find S in this second case.$	$z = - 2, gives S = 1 + 3 \times (- 2) = - 5 z = - 1, gives S = 1 + 3 \times (- 1) = - 2 z = 0, gives S = 1 + 3 \times 0 = 1 z = 1, gives S = 1 + 3 \times 1 = 4 z = 2, gives S = 1 + 3 \times 2 = 7$
$We combine the scores from both cases and see which answer choice does not appear.$	$- 7, - 5, - 4, - 2, - 1, 1, 2, 4, 5, and 7$

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6