At the space station's launch bay, Emek and Tara prepared a cargo capsule for departure. The loading machine showed two formulas for the allowable cargo mass. One screen displayed 5(c + 4) + 2c, while another displayed 4c + 3c + 20, where c represented the number of containers. Tara used the distributive property to rewrite the first expression as 5c + 20 + 2c, which simplified to 7c + 20. Emek combined like terms in the second expression and also obtained 7c + 20. The matching results confirmed that the two formulas were equivalent. With 8 containers scheduled for launch, they evaluated 7(8) + 20 and found an allowable mass of 76 kilograms. The loaded cargo weighed 73 kilograms, so it was within the safe limit. Emek approved the load, and the capsule launched on schedule.

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Challenge 1 of 5

Challenges

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Challenge 1 of 4
Hint
Distribute 5 to both terms inside the parentheses, then combine like terms in each expression.
Which sequence of steps correctly shows that the two cargo formulas 5(c + 4) + 2c and 4c + 3c + 20 are equivalent?
5(c + 4) + 2c = 5c + 4 + 2c = 7c + 4, and 4c + 3c + 20 = 7c + 205(c + 4) + 2c = 5c + 20 + 2c = 7c + 20, and 4c + 3c + 20 = 7c + 205(c + 4) + 2c = 5c + 20 + 2c = 7c + 22, and 4c + 3c + 20 = 7c + 205(c + 4) + 2c = 5c + 20 + 2c = 10c + 20, and 4c + 3c + 20 = 12c + 20
Correct. Both formulas simplify to 7c + 20, so they are equivalent.
Check the distributive property carefully and combine only like terms.
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Hint
First evaluate 7c + 20 when c = 10. Then compare the loaded mass with the allowable mass.
For a later launch, the same cargo rule 7c + 20 is used with 10 containers. If the loaded cargo weighs 84 kilograms, how many kilograms below the allowable mass is the cargo?
Correct. The cargo is 6 kilograms below the allowable mass.
Find the allowable mass for 10 containers first, then subtract the actual cargo mass.
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Hint
Think about the difference between checking one particular value and showing that two expressions have the same simplified form.
Emek says that checking the two original formulas only when c = 8 is enough to prove that they are equivalent for every number of containers. Which statement is correct?
Emek is correct because expressions that match for one value must match for every value.Emek is correct because c can only equal 8 in the cargo formulas.Emek is not correct because equivalent expressions must have different values for some values of c.Emek is not correct because matching for one value does not prove equivalence; simplifying both expressions to 7c + 20 shows they match for all values of c.
Correct. Simplifying both expressions to the same expression shows that they are equivalent for all values of c.
A single matching result checks only one value of c. Consider what simplifying both expressions tells you.
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Kargo Kapsülü Kodu
TR
ARC_G07_PA_02_STORY_02
Kargo Kapsülü Kodu
The Cargo Capsule Code
EN
ARC_G07_PA_02_STORY_02
The Cargo Capsule Code