To analyze the population growth of the two towns represented by the system of equations:
- Equations:
- Town A: \( y = x + 6 \)
- Town B: \( y = 2x + 2 \)
Step 1: Algebraic Solution
To find the point where the two towns' populations are equal, we need to set the two equations equal to each other:
\[ x + 6 = 2x + 2 \]
Step 2: Solve for \(x\)
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Rearranging the equation, we get: \[ 6 - 2 = 2x - x \] \[ 4 = x \]
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Now, substitute \(x = 4\) back into either of the original equations to find \(y\):
Using Town A's equation: \[ y = 4 + 6 = 10 \]
Step 3: Intersection Point
The intersection point of the two lines occurs at: \[ (x, y) = (4, 10) \]
Step 4: Graphical Representation
To graph the equations:
-
For Town A (\(y = x + 6\)):
- When \(x = 0\), \(y = 6\)
- When \(x = -6\), \(y = 0\)
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For Town B (\(y = 2x + 2\)):
- When \(x = 0\), \(y = 2\)
- When \(x = -1\), \(y = 0\)
Plotting
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Plot points for both equations:
- For \(y = x + 6\): (0, 6), (-6, 0)
- For \(y = 2x + 2\): (0, 2), (-1, 0)
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Draw lines through these points.
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Identify the intersection at (4, 10).
Summary
The towns' populations will be equal after 4 years, at which point both towns will have a population of 10.
If you require further analysis or specific graphing instructions, let me know!
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