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Which two equations are equivalent to −3y=6x+4 ?(2 points) Responses y=−2x−43 y is equal to negative 2 x minus 4 thirds −3y=2x+2 negative 3 y is equal to 2 x plus 2 6x+3y=4 6 x plus 3 y is equal to 4 −6y=12x+8 negative 6 y is equal to 12 x plus 8

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To find equations that are equivalent to the given equation \(-3y = 6x + 4\), we can manipulate the equation through algebraic operations.

  1. Rewrite in slope-intercept form (y = mx + b): \(-3y = 6x + 4\) Divide both sides by -3: \[ y = -2x - \frac{4}{3} \] This matches the original equation when rearranged. So \( y = -2x - \frac{4}{3} \) is equivalent.

  2. Check for another equivalent form: If we multiply the entire original equation by -2: \[ 6x + 3y = 4 \] This is another form representing the same relationship.

Now let's look through the options given:

  • Option 1: \(y = -2x - \frac{4}{3}\) (This is equivalent.)
  • Option 2: \(-3y = 2x + 2\) (This is not equivalent since it doesn't match.)
  • Option 3: \(6x + 3y = 4\) (This is equivalent.)
  • Option 4: \(-6y = 12x + 8\) (This simplifies to \(y = -2x - \frac{4}{3}\) by dividing by -6, thus it's equivalent.)

Therefore, the two equations that are equivalent to \(-3y = 6x + 4\) are:

  1. \(y = -2x - \frac{4}{3}\)
  2. \(6x + 3y = 4\)

In summary, the equivalent equations are:

  • \(y = -2x - \frac{4}{3}\)
  • \(6x + 3y = 4\)