During a mission near an unexplored moon, Emek and Tara suddenly lost contact with a research probe. Two tracking stations detected its weak signal, but each provided only a line of possible locations on the coordinate map. Station A reported y = 2x - 3, while Station B reported x + y = 12. Tara used substitution, replacing y in the second equation with 2x - 3. This gave x + 2x - 3 = 12, so 3x = 15 and x = 5. Emek then found y = 7. Before sending the recovery spacecraft, they checked the coordinates carefully. The point (5, 7) satisfied y = 2x - 3, and 5 + 7 also equaled 12. Since both tracking signals agreed at the same point, they sent the spacecraft to (5, 7). Soon afterward, the crew located the missing probe and restored communication.

Challenges

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Challenge 1 of 5

Challenges

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Challenge 1 of 4
Hint
When replacing y, substitute the entire expression that equals y.
A student substitutes y = 2x - 3 into x + y = 12 but writes x + 2x = 12. Which explanation best identifies the student's mistake?
The student should have replaced x with 2x - 3 instead of replacing y.The student should have changed 12 to -12 when substituting.The student should have multiplied the entire equation x + y = 12 by 2.The student omitted the -3 from y = 2x - 3, so the correct substituted equation is x + 2x - 3 = 12.
Correct! Since y = 2x - 3, the complete expression 2x - 3 must replace y.
Not quite. Substitute all of 2x - 3 for y in the second equation.
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Hint
Replace x with 5 in y = 2x - 3 and simplify.
Tara solves the substituted equation and finds x = 5. Using Station A's equation y = 2x - 3, what value of y should Emek obtain?
Correct! Substituting x = 5 gives y = 7.
Try again. Calculate 2(5) - 3.
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Hint
A solution to a system must make both original equations true.
Before sending the recovery spacecraft to (5, 7), Emek and Tara want to verify that both tracking stations identify that point. Which check correctly justifies their decision?
Station A: 7 = 2(5) - 3 is true, but Station B: 5 + 7 = 12 is false, so the point should be rejected.Station A: 7 = 2(5) - 3 is true, and Station B: 5 + 7 = 12 is true, so the point satisfies both equations.Station A: 5 = 2(7) - 3 is true, and Station B: 7 + 12 = 5 is true, so the point satisfies both equations.Only one tracking equation needs to be true for the point to be the intersection.
Correct! The coordinates (5, 7) satisfy both tracking equations, confirming the probe's location.
Not quite. Substitute x = 5 and y = 7 into both original equations and check whether both statements are true.
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Kurtarma Sondasının Koordinatları
TR
ARC_G09_PA_02_STORY_03
Kurtarma Sondasının Koordinatları
The Rescue Probe Coordinates
EN
ARC_G09_PA_02_STORY_03
The Rescue Probe Coordinates