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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

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 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 .

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)
.

1 answer

To understand why the two given systems of equations are equivalent, let's analyze them step by step.

The original system is:

  1. \(2x + 7y = 14\)
  2. \(4x + 5y = 10\)

The second system is:

  1. \(4x + 14y = 28\)
  2. \(8x + 10y = 20\)

We can check if the second system is equivalent to the first one by examining if we can get to the new equations from the original ones through multiplication.

  1. First Equation:

    • If we multiply the first equation \(2x + 7y = 14\) by \(2\), we get: \[ 2(2x + 7y) = 2(14) \implies 4x + 14y = 28 \] This shows that the first equation in the original system produces the first equation in the new system.

  2. Second Equation:

    • If we multiply the second equation \(4x + 5y = 10\) by \(2\), we obtain: \[ 2(4x + 5y) = 2(10) \implies 8x + 10y = 20 \] This shows that the second equation in the original system produces the second equation in the new system.

Thus, we can conclude that both equations in the original system can be scaled by \(2\) to create the equations in the second system. Therefore, the correct response to why the two systems are equivalent is:

"Both equations, \(2x + 7y = 14\) and \(4x + 5y = 10\), can be multiplied by \(2\) to produce the equivalent system of equations."