Question
A system of equations is shown below.
x+y=5
7x+8y=37
Which of the following are valid first steps in an algebraic method to solve the system of equations?
A.
Divide the first equation by –7 and then add the two equations.
B.
Divide the first equation by 56 and then add the two equations.
C.
Multiply the first equation by 56 and then add the two equations.
D.
Multiply the first equation by –7 and then add the two equations.
x+y=5
7x+8y=37
Which of the following are valid first steps in an algebraic method to solve the system of equations?
A.
Divide the first equation by –7 and then add the two equations.
B.
Divide the first equation by 56 and then add the two equations.
C.
Multiply the first equation by 56 and then add the two equations.
D.
Multiply the first equation by –7 and then add the two equations.
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To determine which options are valid first steps to solve the given system of equations, we need to consider the effects of each proposed operation on the equations.
The original equations are:
1. \( x + y = 5 \)
2. \( 7x + 8y = 37 \)
Let's analyze each option:
A. **Divide the first equation by –7 and then add the two equations.**
- Dividing the first equation \( x + y = 5 \) by -7 gives \( -\frac{1}{7}x - \frac{1}{7}y = -\frac{5}{7} \). Adding this to the second equation will not yield a straightforward elimination or solution method. Thus, this is not a valid first step.
B. **Divide the first equation by 56 and then add the two equations.**
- Dividing the first equation \( x + y = 5 \) by 56 gives \( \frac{1}{56}x + \frac{1}{56}y = \frac{5}{56} \). While this operation is mathematically valid, it complicates matters unnecessarily and does not yield a simple method for solving the system. Thus, this is not a practical first step.
C. **Multiply the first equation by 56 and then add the two equations.**
- Multiplying the first equation by 56 gives \( 56x + 56y = 280 \). Adding that to the second equation \( 7x + 8y = 37 \) does not lead to straightforward simplification or elimination. It can also yield large numbers to work with, making it less practical for quick solving. However, since it is mathematically valid but may not be efficient, we consider this option to be less ideal but ultimately a valid step.
D. **Multiply the first equation by –7 and then add the two equations.**
- Multiplying the first equation \( x + y = 5 \) by -7 gives \( -7x - 7y = -35 \). Adding this result to the second equation \( 7x + 8y = 37 \) results in \( (7x - 7x) + (8y - 7y) = 37 - 35 \), which simplifies to \( y = 2 \). This step is valid and efficiently leads to finding a solution.
Therefore, the valid first steps in an algebraic method to solve the system of equations are:
- **C**: Multiply the first equation by 56 and then add the two equations (though it is less practical).
- **D**: Multiply the first equation by -7 and then add the two equations (which is efficient and straightforward).
Thus, options *C* and *D* are valid first steps.
The original equations are:
1. \( x + y = 5 \)
2. \( 7x + 8y = 37 \)
Let's analyze each option:
A. **Divide the first equation by –7 and then add the two equations.**
- Dividing the first equation \( x + y = 5 \) by -7 gives \( -\frac{1}{7}x - \frac{1}{7}y = -\frac{5}{7} \). Adding this to the second equation will not yield a straightforward elimination or solution method. Thus, this is not a valid first step.
B. **Divide the first equation by 56 and then add the two equations.**
- Dividing the first equation \( x + y = 5 \) by 56 gives \( \frac{1}{56}x + \frac{1}{56}y = \frac{5}{56} \). While this operation is mathematically valid, it complicates matters unnecessarily and does not yield a simple method for solving the system. Thus, this is not a practical first step.
C. **Multiply the first equation by 56 and then add the two equations.**
- Multiplying the first equation by 56 gives \( 56x + 56y = 280 \). Adding that to the second equation \( 7x + 8y = 37 \) does not lead to straightforward simplification or elimination. It can also yield large numbers to work with, making it less practical for quick solving. However, since it is mathematically valid but may not be efficient, we consider this option to be less ideal but ultimately a valid step.
D. **Multiply the first equation by –7 and then add the two equations.**
- Multiplying the first equation \( x + y = 5 \) by -7 gives \( -7x - 7y = -35 \). Adding this result to the second equation \( 7x + 8y = 37 \) results in \( (7x - 7x) + (8y - 7y) = 37 - 35 \), which simplifies to \( y = 2 \). This step is valid and efficiently leads to finding a solution.
Therefore, the valid first steps in an algebraic method to solve the system of equations are:
- **C**: Multiply the first equation by 56 and then add the two equations (though it is less practical).
- **D**: Multiply the first equation by -7 and then add the two equations (which is efficient and straightforward).
Thus, options *C* and *D* are valid first steps.
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.