As their spacecraft approached a wide asteroid field, Emek and Tara had to choose between two safe routes around it. The Expression Engine showed the fuel needed for each route, with d representing travel distance in space units. Route A used 3(d + 6) + 2d fuel units, while Route B used 4d + 22 + d. Emek simplified Route A to 5d + 18, and Tara simplified Route B to 5d + 22. They noticed that Route A would use 4 fewer fuel units for any equal travel distance. For the current conditions, both routes had a distance value of 9. Route A required 63 fuel units, while Route B required 67. After checking that both routes were clear of dangerous asteroids, they selected Route A. Their comparison saved fuel, and the spacecraft passed safely around the asteroid field toward its destination.

Challenges

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Challenge 1 of 5

Challenges

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Challenge 1 of 4
Hint
When distributing 3 across d + 6, multiply both terms by 3.
A student simplifies Route A's expression 3(d + 6) + 2d as 5d + 6. What mistake did the student make?
The student should have combined 3d and 2d to get 6d.The student should have subtracted 2d instead of adding it.The student did not multiply 6 by 3 when using the distributive property.The student should have added d and 6 before multiplying.
Correct. Distributing 3 gives 3d + 18, not 3d + 6.
Check how the 3 outside the parentheses must be applied to both terms inside.
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Hint
Evaluate both expressions when d = 15, then find the difference.
For a later trip, both routes have a distance value of 15. Using the simplified expressions 5d + 18 for Route A and 5d + 22 for Route B, how many fewer fuel units would Route A use than Route B?
Correct. Route A uses 4 fewer fuel units.
Substitute 15 for d in both expressions and compare the two results.
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Hint
Compare 5d + 18 and 5d + 22. Notice which parts are the same and which parts differ.
Tara claims that Route A will use exactly 4 fewer fuel units than Route B whenever both routes have the same distance value. Which reasoning best verifies her claim?
Both expressions have 5d, and the constant terms differ by 22 - 18 = 4, so Route B always uses 4 more fuel units.Route A used 63 fuel units when d = 9, so it must always use 63 fuel units.The coefficient of d is 5 in both expressions, so both routes always use the same amount of fuel.The difference is 4 only when d = 9 because changing d changes the difference between the routes.
Correct. The variable terms are identical, while the constants differ by 4, so the difference stays 4 for any equal value of d.
Compare the two simplified expressions term by term and determine what changes when d changes.
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Asteroit Alanından Geçen Bir Rota Seçmek
TR
ARC_G07_PA_02_STORY_03
Asteroit Alanından Geçen Bir Rota Seçmek
Choosing a Route Through the Asteroid Field
EN
ARC_G07_PA_02_STORY_03
Choosing a Route Through the Asteroid Field