To solve the system of equations and understand what the solution means in the context of Pedro and Sam's situation, let's first restate the equations:
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Pedro's distance from Sam's house as a function of time in minutes: \[ Y = -0.5x + 10 \] Here, \(Y\) represents Pedro's distance from Sam's house after \(x\) minutes. Since Pedro starts 10 miles away and is driving towards Sam at a rate of 0.5 miles per minute, the equation appropriately reflects that with a negative slope.
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Sam's distance from his house as a function of time in minutes: \[ Y = 0.75x \] In this equation, \(Y\) represents the distance Sam has traveled from his house after \(x\) minutes. Since he is driving towards Pedro at a rate of 0.75 miles per minute, this equation has a positive slope.
To find the point where they meet, we can set the two equations equal to each other:
\[ -0.5x + 10 = 0.75x \]
Now, let's solve for \(x\):
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Combine like terms: \[ 10 = 0.75x + 0.5x \]
\[ 10 = 1.25x \]
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Solve for \(x\): \[ x = \frac{10}{1.25} = 8 \]
Now that we have \(x = 8\), we can find the corresponding \(Y\) value to determine how far each person has traveled by substituting \(x\) back into either equation. Let's use Sam's equation:
\[ Y = 0.75(8) = 6 \]
Thus, the solution to the system is \( (8, 6) \).
Interpretation of the Solution:
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Time: The result indicates that they will meet after 8 minutes.
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Distance: The value of \(Y\) (6 miles) shows how far Sam has traveled from his house when they meet. This also implies that Pedro has 4 miles left to go to meet Sam (because 10 - 6 = 4).
Conclusion:
In this context, the solution means that after 8 minutes of driving, Pedro and Sam will meet each other after Pedro has traveled 4 miles towards Sam's house, while Sam has driven 6 miles from his house towards Pedro.
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