Question
How do you use multiplication in elimination to show that this system of equations has infinite solutions.
4x - 8y = -20
-8x + 16y = 40
A. Multiply the first equation by 4.
B. Multiply the first equation by -2 and both equations will be the same.
C. Multiply the second equation by 2 and both equations will be the same.
D. Multiply the two equations together.
4x - 8y = -20
-8x + 16y = 40
A. Multiply the first equation by 4.
B. Multiply the first equation by -2 and both equations will be the same.
C. Multiply the second equation by 2 and both equations will be the same.
D. Multiply the two equations together.
Answers
GPT-4o mini
To show that the system of equations has infinite solutions using multiplication in elimination, we need to manipulate the equations so that they are effectively the same.
The system of equations is:
1. \(4x - 8y = -20\)
2. \(-8x + 16y = 40\)
Let's analyze the options:
A. **Multiply the first equation by 4.**
- If we multiply \(4x - 8y = -20\) by 4, we get \(16x - 32y = -80\), which does not help in showing that the two equations are equivalent.
B. **Multiply the first equation by -2 and both equations will be the same.**
- Multiplying the first equation \(4x - 8y = -20\) by -2 results in \(-8x + 16y = 40\), which is exactly the same as the second equation. This demonstrates that the two equations are indeed equivalent. So this option shows that there are infinite solutions.
C. **Multiply the second equation by 2 and both equations will be the same.**
- If we multiply the second equation \(-8x + 16y = 40\) by 2, we get \(-16x + 32y = 80\), which does not help in establishing equivalence with the first equation.
D. **Multiply the two equations together.**
- This option does not make sense in the context of elimination or determining equivalence.
The correct answer is **B. Multiply the first equation by -2 and both equations will be the same.** This shows that there are infinite solutions to the system because the two equations represent the same line.
The system of equations is:
1. \(4x - 8y = -20\)
2. \(-8x + 16y = 40\)
Let's analyze the options:
A. **Multiply the first equation by 4.**
- If we multiply \(4x - 8y = -20\) by 4, we get \(16x - 32y = -80\), which does not help in showing that the two equations are equivalent.
B. **Multiply the first equation by -2 and both equations will be the same.**
- Multiplying the first equation \(4x - 8y = -20\) by -2 results in \(-8x + 16y = 40\), which is exactly the same as the second equation. This demonstrates that the two equations are indeed equivalent. So this option shows that there are infinite solutions.
C. **Multiply the second equation by 2 and both equations will be the same.**
- If we multiply the second equation \(-8x + 16y = 40\) by 2, we get \(-16x + 32y = 80\), which does not help in establishing equivalence with the first equation.
D. **Multiply the two equations together.**
- This option does not make sense in the context of elimination or determining equivalence.
The correct answer is **B. Multiply the first equation by -2 and both equations will be the same.** This shows that there are infinite solutions to the system because the two equations represent the same line.