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Home » Let \(v = u^2\), so \(v \in (1, 2]\). Then: - Londonproperty

Let \(v = u^2\), so \(v \in (1, 2]\). Then: - Londonproperty

📅 March 13, 2026 👤 scraface

Mar 13, 2026

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📌 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?

📌 Solution: Let $ l $ be the cost of a software license and $ h $ the cost of a hardware upgrade. The system is:

📌 7l + 5h = 120.50 \\

📌 4l + 3h = 85.00

📌 Multiply the first equation by 3: $ 21l + 15h = 361.50 $. Multiply the second equation by 5: $ 20l + 15h = 425.00 $. Subtract the first modified equation from the second: $ (20l + 15h) - (21l + 15h) = 425.00 - 361.50 \Rightarrow -l = 63.50 \Rightarrow l = -63.50 $. This negative value indicates an error in assumptions, suggesting a possible misinterpretation of the problem. However, if the equations are correct, the solution is $ \boxed{-63.50} $, which is invalid in context. Rechecking the problem setup is advised.

📌 Question: If $ f(x) = 3x - 7 $ and $ g(x) = x^2 + 2x $, what is $ g(f(4)) $?

📌 Solution: First compute $ f(4) = 3(4) - 7 = 12 - 7 = 5 $. Then substitute into $ g(x) $: $ g(5) = 5^2 + 2(5) = 25 + 10 = 35 $. The value of $ g(f(4)) $ is $ \boxed{35} $.

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