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 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) . All equations share the point left parenthesis 0 comma 2 right parenthesis . 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.
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 \(8x + 10y = 20\), let's analyze the equations step by step.
1. **Multiplying Equations**: The first system can be transformed into the second system by multiplying each equation by a certain factor:
- Multiply the first equation \(2x + 7y = 14\) by \(2\):
\[
2(2x + 7y) = 2 \times 14 \Rightarrow 4x + 14y = 28
\]
- Multiply the second equation \(4x + 5y = 10\) by \(2\):
\[
2(4x + 5y) = 2 \times 10 \Rightarrow 8x + 10y = 20
\]
2. **Conclusion**: Both equations in the first system can be multiplied by \(2\) to produce the second system. This means that the two systems of equations are equivalent, as they represent the same lines in the coordinate plane.
Looking at the provided responses, the correct one is that both equations \(2x + 7y = 14\) and \(4x + 5y = 10\) can be multiplied by \(2\) to produce the equivalent system of equations \(4x + 14y = 28\) and \(8x + 10y = 20\).
So the correct response is that both equations can be multiplied by a constant to form the equivalent system.
1. **Multiplying Equations**: The first system can be transformed into the second system by multiplying each equation by a certain factor:
- Multiply the first equation \(2x + 7y = 14\) by \(2\):
\[
2(2x + 7y) = 2 \times 14 \Rightarrow 4x + 14y = 28
\]
- Multiply the second equation \(4x + 5y = 10\) by \(2\):
\[
2(4x + 5y) = 2 \times 10 \Rightarrow 8x + 10y = 20
\]
2. **Conclusion**: Both equations in the first system can be multiplied by \(2\) to produce the second system. This means that the two systems of equations are equivalent, as they represent the same lines in the coordinate plane.
Looking at the provided responses, the correct one is that both equations \(2x + 7y = 14\) and \(4x + 5y = 10\) can be multiplied by \(2\) to produce the equivalent system of equations \(4x + 14y = 28\) and \(8x + 10y = 20\).
So the correct response is that both equations can be multiplied by a constant to form the equivalent system.