Final answer:
Using the addition method, the solutions for the systems of equations a) and b) are x = 2, y = 3 and x = -6/5, y = 7/5 respectively. None of the provided answer choices match these solutions, indicating a possible typo or incorrect choices given.
Step-by-step explanation:
Solve the Systems by the Addition Method
To solve the given systems of equations using the addition (or elimination) method, we manipulate the equations to eliminate one variable and solve for the other. Let's tackle each system separately.
a) System 1
The first system of equations is:
We will multiply the second equation by 2 so that we can eliminate y when we add the equations together.
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- 2*(2x + y = 7) → 4x + 2y = 14
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- Add: (x - 2y) + (4x + 2y) = -4 + 14
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- Simplify: 5x = 10
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- Divide by 5: x = 2
Now, we substitute x back into one of the original equations to find y.
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- Substitute x in the second equation: 2*(2) + y = 7
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- Simplify: 4 + y = 7
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- Solve for y: y = 7 - 4
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- y = 3
Thus, the solution for the first system is x = 2 and y = 3.
b) System 2
The second system of equations is:
We will multiply the first equation by 2 to match the coefficient of y in the second equation.
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- 2*(2x + y = -1) → 4x + 2y = -2
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- Now add this to the second equation: (4x + 2y) + (x - 2y) = -2 + (-4)
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- Simplify: 5x = -6
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- Divide by 5: x = -6/5
Once again, substitute x back into one of the original equations to solve for y.
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- Substitute x in the second equation: 2*(-6/5) + y = -1
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- Multiply: -12/5 + y = -1
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- Convert -1 to fraction: -5/5
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- Add 12/5 to both sides: y = -5/5 + 12/5
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- y = 7/5
The solution for the second system is x = -6/5 and y = 7/5 or x = -1.2 and y = 1.4 when converted to decimal form.
After examining both solutions, it is clear that none of the provided choices (A, B, C, or D) match the solutions we obtained. Therefore, there may be a typo in the initial question, or none of the choices are correct.