Question
Serina wants to solve the following system of equations in the most efficient way.
2 x + 3 y = 18. x + 7 y = 31.
She plans to solve for x in the first equation as her first step since both 2 and 3 can be divided into 18. Why is Serina mistaken?
Serina should have solved for x in the second equation because it has a coefficient of 1.
Serina should have solved for y in the first equation because dividing by 3 instead of by 2 would give a smaller number in the solution.
Serina should have solved for y in the second equation because it has the largest coefficient.
Serina should have solved for y in the first equation because the division step will be easier since 18 is divisible by 3.
2 x + 3 y = 18. x + 7 y = 31.
She plans to solve for x in the first equation as her first step since both 2 and 3 can be divided into 18. Why is Serina mistaken?
Serina should have solved for x in the second equation because it has a coefficient of 1.
Serina should have solved for y in the first equation because dividing by 3 instead of by 2 would give a smaller number in the solution.
Serina should have solved for y in the second equation because it has the largest coefficient.
Serina should have solved for y in the first equation because the division step will be easier since 18 is divisible by 3.
Answers
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Answered by
GPT-4o mini
Serina's reasoning about solving the equations is flawed because she focuses solely on the coefficients and the constant term. Let's analyze her choices:
1. **Solving for x in the first equation (2x + 3y = 18)**: This approach is not inherently mistaken, but it may not be the most efficient given that the coefficient of \(x\) is 2 (you would have to deal with fractions or division by 2).
2. **Solving for x in the second equation (x + 7y = 31)**: This could be considered a more efficient approach because 1 (the coefficient of \(x\)) is much simpler to work with compared to 2.
3. **Solving for y in the first equation**: This could also be an option. However, the choice to divide by 3 is not primarily a limitation if you want to isolate \(y\).
4. **Solving for y in the second equation**: This option has little merit in this context because while it has a larger coefficient, the result of solving normally does not relate directly to efficiency.
5. **Solving for y in the first equation, reasoning that division will be easier since 18 is divisible by 3**: This choice suggests that it’s easier to get to a solution. The division by 3 could indeed make calculations simpler.
Considering these points, Serina's mistake isn’t about whether her choice is technically solvable but rather choosing a method (solving for \(x\) in the first equation) that could lead to a more complicated path toward the solution than necessary.
The best clarification based on the provided options is:
Serina should have solved for **x in the second equation because it has a coefficient of 1.**
This simplifies calculations and leads to a more straightforward path to the solution.
1. **Solving for x in the first equation (2x + 3y = 18)**: This approach is not inherently mistaken, but it may not be the most efficient given that the coefficient of \(x\) is 2 (you would have to deal with fractions or division by 2).
2. **Solving for x in the second equation (x + 7y = 31)**: This could be considered a more efficient approach because 1 (the coefficient of \(x\)) is much simpler to work with compared to 2.
3. **Solving for y in the first equation**: This could also be an option. However, the choice to divide by 3 is not primarily a limitation if you want to isolate \(y\).
4. **Solving for y in the second equation**: This option has little merit in this context because while it has a larger coefficient, the result of solving normally does not relate directly to efficiency.
5. **Solving for y in the first equation, reasoning that division will be easier since 18 is divisible by 3**: This choice suggests that it’s easier to get to a solution. The division by 3 could indeed make calculations simpler.
Considering these points, Serina's mistake isn’t about whether her choice is technically solvable but rather choosing a method (solving for \(x\) in the first equation) that could lead to a more complicated path toward the solution than necessary.
The best clarification based on the provided options is:
Serina should have solved for **x in the second equation because it has a coefficient of 1.**
This simplifies calculations and leads to a more straightforward path to the solution.
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