Question
Why is the system of equations 2x+7y=14 and 4x+5y=10 equivalent to the system of equations 4x+14y=28 and 8x+10y=20 ?
A: The intercepts of both 4x + 5y = 10 and 8x + 10y = 20 are ( 0, 2.5) and ( 2, 0)
B: Both equations, 2x + 7y = 14 and 4x + 5y = 10 can be multiplied by 1/2 to produce the equivalent system of equations.
C: All equations share the point ( 0,2)
D: The intercepts of the equations 2x + 7y = 14 and 4x + 14y = 28 are ( 0,2 ) and ( 7, 0)
A: The intercepts of both 4x + 5y = 10 and 8x + 10y = 20 are ( 0, 2.5) and ( 2, 0)
B: Both equations, 2x + 7y = 14 and 4x + 5y = 10 can be multiplied by 1/2 to produce the equivalent system of equations.
C: All equations share the point ( 0,2)
D: The intercepts of the equations 2x + 7y = 14 and 4x + 14y = 28 are ( 0,2 ) and ( 7, 0)
Answers
GPT-4o mini
To determine why the system of equations \( 2x + 7y = 14 \) and \( 4x + 5y = 10 \) is equivalent to the system \( 4x + 14y = 28 \) and \( 8x + 10y = 20 \), we need to analyze the relationships between these equations.
1. **Identify if the equations are equivalent**:
- The first equation \( 2x + 7y = 14 \) can be multiplied by 2 to give \( 4x + 14y = 28 \).
- The second equation \( 4x + 5y = 10 \) can be multiplied by 2 to give \( 8x + 10y = 20 \).
Since both equations in the original system yield the new equations in the second system when multiplied by a constant, the two systems are equivalent.
2. **Evaluate the answer choices**:
- **Choice A**: States something about the intercepts that is irrelevant to determining equivalence.
- **Choice B**: Correctly identifies that both original equations can be multiplied by a factor to produce the new equations, which makes them equivalent.
- **Choice C**: Mentions a specific point; however, this does not confirm equivalence.
- **Choice D**: Also mentions intercepts and doesn't directly address the equivalence clearly.
Based on this analysis, the correct choice is:
**B: Both equations, 2x + 7y = 14 and 4x + 5y = 10 can be multiplied by 1/2 to produce the equivalent system of equations.**
However, please note that the actual correct factor needed should be multiplied by 2 rather than 1/2, just to clarify the multiplication and equivalence relationship. Alternatively, a clearer statement would specify the factor needed for each equation.
1. **Identify if the equations are equivalent**:
- The first equation \( 2x + 7y = 14 \) can be multiplied by 2 to give \( 4x + 14y = 28 \).
- The second equation \( 4x + 5y = 10 \) can be multiplied by 2 to give \( 8x + 10y = 20 \).
Since both equations in the original system yield the new equations in the second system when multiplied by a constant, the two systems are equivalent.
2. **Evaluate the answer choices**:
- **Choice A**: States something about the intercepts that is irrelevant to determining equivalence.
- **Choice B**: Correctly identifies that both original equations can be multiplied by a factor to produce the new equations, which makes them equivalent.
- **Choice C**: Mentions a specific point; however, this does not confirm equivalence.
- **Choice D**: Also mentions intercepts and doesn't directly address the equivalence clearly.
Based on this analysis, the correct choice is:
**B: Both equations, 2x + 7y = 14 and 4x + 5y = 10 can be multiplied by 1/2 to produce the equivalent system of equations.**
However, please note that the actual correct factor needed should be multiplied by 2 rather than 1/2, just to clarify the multiplication and equivalence relationship. Alternatively, a clearer statement would specify the factor needed for each equation.