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Which of the following systems of equations is equivalent to the system given below?4x+9y=102x+3y=12(1 point)Responses4x+9y=104x+3y=244 x plus 9 y is equal to 10 4 x plus 3 y is equal to 244x+9y=102x+9y=364 x plus 9 y is equal to 10 2 x plus 9 y is equal to 364x+9y=104x+6y=244 x plus 9 y is equal to 10 4 x plus 6 y is equal to 244x+9y=102x+9y=12

1 answer

To find an equivalent system of equations, we can manipulate and simplify the original system, which consists of the following equations:

  1. \(4x + 9y = 10\)
  2. \(2x + 3y = 12\)

Let's look at the second equation, \(2x + 3y = 12\). If we multiply this equation by 2, it will provide us with a new equation that is still equivalent to the second equation:

\[ 2(2x + 3y) = 2(12) \ 4x + 6y = 24 \]

Now we have a new equivalent system:

  1. \(4x + 9y = 10\) (unchanged)
  2. \(4x + 6y = 24\) (transformed from the original second equation)

Now let’s see which of the responses matches this system.

  1. Response: \(4x + 9y = 10\) and \(4x + 3y = 24\) (not matched)
  2. Response: \(4x + 9y = 10\) and \(2x + 9y = 36\) (not matched)
  3. Response: \(4x + 9y = 10\) and \(2x + 3y = 12\) (match original, but not manipulated)
  4. Response: \(4x + 9y = 10\) and \(4x + 6y = 24\) (this is a match)
  5. Response: \(4x + 9y = 10\) and \(4x + 6y = 24\) (this is a match)

Therefore, the equivalent system of equations is:

\[ \text{System: } 4x + 9y = 10 \text{ and } 4x + 6y = 24 \]

So, the correct choice is:

Response: \(4x + 9y = 10\) and \(4x + 6y = 24\)