Let's examine Amos's solution step by step and identify any mistakes.
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Starting with the equations:
- Equation 1: \( 8x + 5y = 18 \)
- Equation 2: \( 6x + y = -2 \)
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Amos correctly decided to multiply the second equation by -5: \[ -5(6x + y) = -5(-2) \implies -30x - 5y = 10 \]
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Now we have:
- Equation 1: \( 8x + 5y = 18 \)
- Modified Equation 2: \( -30x - 5y = 10 \)
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Next, when he added the two equations to eliminate \(y\): \[ (8x + 5y) + (-30x - 5y) = 18 + 10 \implies -22x = 28 \] Therefore, simplifying gives: \[ -22x = 28 \implies x = \frac{28}{-22} \implies x = -\frac{14}{11} \quad (\text{or approximately } -1.273) \]
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Now, substituting \( x = 2 \) into: \[ 6x + y = -2 \implies 6(2) + y = -2 \implies 12 + y = -2 \implies y = -2 - 12 = -14 \]
However, this substitution assumes \( x = 2 \) was correct, but we've already shown that the calculation for \(x\) was incorrect.
To summarize:
- The error occurred when Amos added the equations. He should have derived \( -22x = 28 \) from the addition, which indicates a calculation mistake in the use of coefficients, leading to the false conclusion that \( x = 2 \).
The final answer: He solved the equation incorrectly when trying to eliminate \(y\) with the incorrect addition of the equations.