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?(1 point) Responses The intercepts of the equations 2x+7y=14 and 4x+14y=28 are (0,2) and (7,0). The intercepts of the equations 2 x plus 7 y equals 14 and 4 x plus 14 y equals 28 are left parenthesis 0 comma 2 right parenthesis and left parenthesis 7 comma 0 right parenthesis . Both equations, 2x+7y=14 and 4x+5y=10, can be multiplied by 12 to produce the equivalent system of equations. Both equations, 2 x plus 7 y equals 14 and 4 x plus 5 y equals 10 , can be multiplied by Start Fraction 1 over 2 End Fraction to produce the equivalent system of equations. The intercepts of both 4x+5y=10 and 8x+10y=20 are (0,2.5) and (2,0). The intercepts of both 4 x plus 5 y equals 10 and 8 x plus 10 y equals 20 are left parenthesis 0 comma 2.5 right parenthesis and left parenthesis 2 comma 0 right parenthesis . All equations share the point (0,2).
Answers
Answer
THATS NOT IN THE GIVEN OPTIONS IDIOT
Answered by
GPT-4o mini
To determine whether the systems of equations are equivalent, we should focus on the concept of equivalent systems. Two systems of equations are considered equivalent if they have the same solution set, which can often be demonstrated by manipulating one or both of the systems (for example, scaling equations).
The first system is:
1. \(2x + 7y = 14\)
2. \(4x + 5y = 10\)
The second system is:
3. \(4x + 14y = 28\)
4. \(8x + 10y = 20\)
Let's check if we can manipulate the first system to obtain the second one.
- If we multiply the first equation \(2x + 7y = 14\) by 2, we get:
\[
4x + 14y = 28
\]
(This matches the first equation of the second system.)
- If we multiply the second equation \(4x + 5y = 10\) by 2, we have:
\[
8x + 10y = 20
\]
(This matches the second equation of the second system.)
Since we can obtain the second system from the first by multiplying each equation by 2, this shows that the two systems are indeed equivalent.
### Conclusion:
Both equations in the first system can be multiplied by \(2\) to produce the equivalent system of equations, hence the correct response is:
**Both equations, \(2x + 7y = 14\) and \(4x + 5y = 10\), can be multiplied by 2 to produce the equivalent system of equations.**
The first system is:
1. \(2x + 7y = 14\)
2. \(4x + 5y = 10\)
The second system is:
3. \(4x + 14y = 28\)
4. \(8x + 10y = 20\)
Let's check if we can manipulate the first system to obtain the second one.
- If we multiply the first equation \(2x + 7y = 14\) by 2, we get:
\[
4x + 14y = 28
\]
(This matches the first equation of the second system.)
- If we multiply the second equation \(4x + 5y = 10\) by 2, we have:
\[
8x + 10y = 20
\]
(This matches the second equation of the second system.)
Since we can obtain the second system from the first by multiplying each equation by 2, this shows that the two systems are indeed equivalent.
### Conclusion:
Both equations in the first system can be multiplied by \(2\) to produce the equivalent system of equations, hence the correct response is:
**Both equations, \(2x + 7y = 14\) and \(4x + 5y = 10\), can be multiplied by 2 to produce the equivalent system of equations.**
Answered by
GPT-4o mini
I apologize for the oversight. Let's analyze your provided options and see which one is the correct explanation for why the two systems of equations are equivalent.
1. **The intercepts of the equations 2x + 7y = 14 and 4x + 14y = 28 are (0,2) and (7,0).**
- This statement is incorrect; it mentions only one equation in each case and seems to confuse the intercepts.
2. **Both equations, 2x + 7y = 14 and 4x + 14y = 28, can be multiplied by 12 to produce the equivalent system of equations.**
- This statement is incorrect; multiplying by 12 would not produce the second system.
3. **Both equations, 2x + 7y = 14 and 4x + 5y = 10, can be multiplied by \( \frac{1}{2} \) to produce the equivalent system of equations.**
- This statement is also incorrect as multiplying these equations by \( \frac{1}{2} \) would not yield the second system.
4. **The intercepts of both 4x + 5y = 10 and 8x + 10y = 20 are (0,2.5) and (2,0).**
- This statement is only correct for \(4x + 5y = 10\) but incorrect for \(8x + 10y = 20\).
5. **All equations share the point (0,2).**
- This seems plausible because both systems do include equations that may have common points; however, checking the actual solution sets is necessary.
Based on the analysis:
- The only option that indicates any truth is that the equations can indeed represent the same line when manipulated, particularly through scaling, but none of the provided statements accurately summarizes that understanding.
Given the choices you’ve provided, **none of the options are entirely correct or relevant for identifying the equivalence** between the two systems of equations. If you can clarify or adjust the options, I could provide a more accurate assessment. Again, I’m sorry for the confusion earlier!
1. **The intercepts of the equations 2x + 7y = 14 and 4x + 14y = 28 are (0,2) and (7,0).**
- This statement is incorrect; it mentions only one equation in each case and seems to confuse the intercepts.
2. **Both equations, 2x + 7y = 14 and 4x + 14y = 28, can be multiplied by 12 to produce the equivalent system of equations.**
- This statement is incorrect; multiplying by 12 would not produce the second system.
3. **Both equations, 2x + 7y = 14 and 4x + 5y = 10, can be multiplied by \( \frac{1}{2} \) to produce the equivalent system of equations.**
- This statement is also incorrect as multiplying these equations by \( \frac{1}{2} \) would not yield the second system.
4. **The intercepts of both 4x + 5y = 10 and 8x + 10y = 20 are (0,2.5) and (2,0).**
- This statement is only correct for \(4x + 5y = 10\) but incorrect for \(8x + 10y = 20\).
5. **All equations share the point (0,2).**
- This seems plausible because both systems do include equations that may have common points; however, checking the actual solution sets is necessary.
Based on the analysis:
- The only option that indicates any truth is that the equations can indeed represent the same line when manipulated, particularly through scaling, but none of the provided statements accurately summarizes that understanding.
Given the choices you’ve provided, **none of the options are entirely correct or relevant for identifying the equivalence** between the two systems of equations. If you can clarify or adjust the options, I could provide a more accurate assessment. Again, I’m sorry for the confusion earlier!
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