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Which of the following shows a system of equations equivalent to 2x+7y=14 and 4x+5y=10?

3 answers

To find a system of equations equivalent to the given equations \(2x + 7y = 14\) and \(4x + 5y = 10\), we can manipulate these equations through operations like multiplication or addition.

  1. Start with the given equations: \[ 2x + 7y = 14 \tag{1} \] \[ 4x + 5y = 10 \tag{2} \]

  2. Multiply equation (1) by 2: \[ 2(2x + 7y) = 2(14) \] \[ 4x + 14y = 28 \tag{3} \]

  3. Now we have the following system of equations: \[ 4x + 14y = 28 \tag{3} \] \[ 4x + 5y = 10 \tag{2} \]

This new set of equations is equivalent to the original equations, as they represent the same lines in the coordinate plane.

If you have multiple options to consider, look for a system that includes \(4x + 14y = 28\) and \(4x + 5y = 10\) or any other equivalent forms derived from scaling or combining the original equations.

Let me know if you have any specific options in mind, and I can help you check their equivalence!

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 ?

To determine why the system of equations \(2x + 7y = 14\) and \(4x + 5y = 10\) is equivalent to \(4x + 14y = 28\) and \(8x + 10y = 20\), we need to check if both sets of equations represent the same relationships.

  1. Starting with the original equations: \[ 2x + 7y = 14 \tag{1} \] \[ 4x + 5y = 10 \tag{2} \]

  2. Multiply equation (1) by 2 to compare with a transformed version: \[ 2(2x + 7y) = 2(14) \] This gives us: \[ 4x + 14y = 28 \tag{3} \]

    So the first equation in the new system \(4x + 14y = 28\) is indeed equivalent to \(2x + 7y = 14\).

  3. Now, let's examine equation (2): Multiply \(4x + 5y = 10\) by 2: \[ 2(4x + 5y) = 2(10) \] This gives us: \[ 8x + 10y = 20 \tag{4} \]

    Thus, the second equation \(8x + 10y = 20\) is equivalent to \(4x + 5y = 10\).

Combining the transformed equations, we have:

  • From (\(2x + 7y = 14\)), we derived \(4x + 14y = 28\).
  • From (\(4x + 5y = 10\)), we derived \(8x + 10y = 20\).
  1. Conclusion: Since both equations \(4x + 14y = 28\) and \(8x + 10y = 20\) were derived from the original system of equations, the systems of equations are equivalent. They describe the same lines with the same solutions in terms of \(x\) and \(y\).

Hence, the system of equations \(2x + 7y = 14\) and \(4x + 5y = 10\) is equivalent to the system \(4x + 14y = 28\) and \(8x + 10y = 20\).