Aboard a space station, Emek and Tara prepared to distribute a new shipment, but part of its digital record had been damaged. They needed to determine the numbers of food containers, f, and equipment containers, e. One surviving panel showed that two recorded groups contained 2f + 3e = 72 containers in total. Another panel gave the relationship 3f + 2e = 68. Tara chose elimination because the coefficients could be matched easily. She multiplied the first equation by 3 and the second by 2, obtaining 6f + 9e = 216 and 6f + 4e = 136. Subtracting the equations gave 5e = 80, so e = 16. Substituting 16 into 3f + 2e = 68 gave f = 12. Emek checked both original equations and confirmed the results. With 12 food containers and 16 equipment containers identified, they safely distributed the shipment to the correct storage areas.

Challenges

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Challenge 1 of 5

Challenges

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Challenge 1 of 4

Hint

Find a common multiple of the coefficients 2 and 3 in front of f.

Tara wants to eliminate f from the system 2f + 3e = 72 and 3f + 2e = 68. Which pair of operations will give the same coefficient for f in both equations?

Multiply the first equation by 2 and the second equation by 3.Multiply the first equation by 3 and the second equation by 2.Multiply both equations by 2.Multiply both equations by 3.

Correct! These operations make both f coefficients equal to 6, so f can be eliminated.

Not quite. Choose multipliers that make the coefficients of f equal in the two equations.

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Hint

Subtract corresponding terms. The f terms will cancel, leaving an equation involving only e.

After Tara multiplies the equations, she has 6f + 9e = 216 and 6f + 4e = 136. Subtract the second equation from the first. What is the resulting value of e?

Correct! Subtraction gives 5e = 80, so e = 16.

Try again. Subtract the second equation from the first, then solve the resulting equation for e.

🎉 Congrats! You got this question!

Hint

Replace e with 16, simplify 2(16), and then isolate f.

Tara finds e = 16. Which calculation correctly uses this result with 3f + 2e = 68 to determine the number of food containers?

3f + 2(16) = 68, so 3f = 36 and f = 12.3f + 16 = 68, so 3f = 52 and f = 52/3.3f + 2(16) = 68, so 3f = 100 and f = 100/3.3f + 2(16) = 68, so 3f = 34 and f = 34/3.

Correct! Substituting e = 16 gives 3f + 32 = 68, so f = 12.

Not quite. Substitute 16 for e, multiply it by its coefficient 2, and then solve for f.

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