Question
To solve the system of linear equations 8 x + 5 y = 18 and 6 x + y = negative 2. by using the linear combination method, Amos decided that he should first multiply the second equation by –5 and then add the two equations together to eliminate the y-terms. His calculations are as shown.
8 x + 5 y = 18. + 6 x minus 5 y = 10. 14 x = 28. StartFraction 14 x Over 14 EndFraction = StartFraction 28 Over 14 EndFraction. X = 2.
6 x + y = negative 2. 6 (2) + y = negative 2. 12 + y = negative 2. 12 + y minus 12 = negative 2 minus 12. y = negative 14.
Amos’s solution is (2, –14). What did he do wrong?
He multiplied the equation 6 x + y = negative 2. by –5 incorrectly.
He added the equations 8 x + 5 y = 18 and 8 x + 5 y = 18 incorrectly.
He substituted 2 into the equation 6 x + y = negative 2. incorrectly.
He solved the equation 12 + y = negative 2 for y incorrectly.
8 x + 5 y = 18. + 6 x minus 5 y = 10. 14 x = 28. StartFraction 14 x Over 14 EndFraction = StartFraction 28 Over 14 EndFraction. X = 2.
6 x + y = negative 2. 6 (2) + y = negative 2. 12 + y = negative 2. 12 + y minus 12 = negative 2 minus 12. y = negative 14.
Amos’s solution is (2, –14). What did he do wrong?
He multiplied the equation 6 x + y = negative 2. by –5 incorrectly.
He added the equations 8 x + 5 y = 18 and 8 x + 5 y = 18 incorrectly.
He substituted 2 into the equation 6 x + y = negative 2. incorrectly.
He solved the equation 12 + y = negative 2 for y incorrectly.
Answers
Answered by
GPT-4o mini
Let's examine Amos's solution step by step and identify any mistakes.
1. Starting with the equations:
- Equation 1: \( 8x + 5y = 18 \)
- Equation 2: \( 6x + y = -2 \)
2. Amos correctly decided to multiply the second equation by -5:
\[
-5(6x + y) = -5(-2) \implies -30x - 5y = 10
\]
3. Now we have:
- Equation 1: \( 8x + 5y = 18 \)
- Modified Equation 2: \( -30x - 5y = 10 \)
4. Next, when he added the two equations to eliminate \(y\):
\[
(8x + 5y) + (-30x - 5y) = 18 + 10 \implies -22x = 28
\]
Therefore, simplifying gives:
\[
-22x = 28 \implies x = \frac{28}{-22} \implies x = -\frac{14}{11} \quad (\text{or approximately } -1.273)
\]
5. 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.
1. Starting with the equations:
- Equation 1: \( 8x + 5y = 18 \)
- Equation 2: \( 6x + y = -2 \)
2. Amos correctly decided to multiply the second equation by -5:
\[
-5(6x + y) = -5(-2) \implies -30x - 5y = 10
\]
3. Now we have:
- Equation 1: \( 8x + 5y = 18 \)
- Modified Equation 2: \( -30x - 5y = 10 \)
4. Next, when he added the two equations to eliminate \(y\):
\[
(8x + 5y) + (-30x - 5y) = 18 + 10 \implies -22x = 28
\]
Therefore, simplifying gives:
\[
-22x = 28 \implies x = \frac{28}{-22} \implies x = -\frac{14}{11} \quad (\text{or approximately } -1.273)
\]
5. 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.
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