5p + 2s = 48.50 \\

February 22, 2026

$5p + 2s = 48.50 \\$

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Multiply the second equation by 3: $ 9x + 6y = 54.90 $. Subtract the first equation: $ (9x + 6y) - (4x + 6y) = 54.90 - 32.40 $, yielding $ 5x = 22.50 $. Thus, $ x = 4.50 $ dollars, or $ \boxed{450} $ cents.
Question: An archaeologist discovers 5 pottery shards and 2 stone tools worth \$48.50, and 3 pottery shards and 4 stone tools worth \$51.00. What is the value of one stone tool in dollars?
Solution: Let $ p $ be the price of a pottery shard and $ s $ the price of a stone tool. The system is:
3p + 4s = 51.00
Multiply the first equation by 2: $ 10p + 4s = 97.00 $. Subtract the second equation: $ (10p + 4s) - (3p + 4s) = 97.00 - 51.00 $, yielding $ 7p = 46.00 $. Thus, $ p = \frac{46}{7} \approx 6.57 $. Substitute into the first equation: $ 5(6.57) + 2s = 48.50 \Rightarrow 32.85 + 2s = 48.50 \Rightarrow 2s = 15.65 \Rightarrow s = 7.825 $. The value of a stone tool is $ \boxed{7.83} $ dollars.
Question: A researcher spends \$120.50 on 7 software licenses and 5 hardware upgrades, and \$85.00 on 4 software licenses and 3 hardware upgrades. What is the cost of a software license?

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