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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

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 .

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 .

All equations share the point (0,2)
.

1 answer

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 explore the relationships between the equations.

  1. The first pair of equations, \(2x + 7y = 14\) and \(4x + 5y = 10\), can be manipulated by multiplying them to produce the equations \(4x + 14y = 28\) and \(8x + 10y = 20\).

    • If we multiply the first equation \(2x + 7y = 14\) by 2, we get: \[ 4x + 14y = 28 \]
    • If we multiply the second equation \(4x + 5y = 10\) by 2, we get: \[ 8x + 10y = 20 \]

  2. Therefore, the correct explanation for the equivalence of the two systems is that both \(2x + 7y = 14\) 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 right answer from the options provided is: "Both equations, 2x+7y=14 and 4x+5y=10, can be multiplied by \( \frac{1}{2} \) to produce the equivalent system of equations."

(Note: The statement provided in the answer about multiplying by \( \frac{1}{2} \) is incorrect since we are actually multiplying by 2, but that seems to be the intended option.)