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Choose the two equations that are equivalent to 4x+y=3 .

CHOOSE TWO.

(2 points)
Responses

y=−4x+3y is equal to negative 4 x plus 3

x+4y=3x plus 4 y is equal to 3

8x+2y=68 x plus 2 y is equal to 6

3x+y=4

1 answer

To find the equations equivalent to \(4x + y = 3\), we can manipulate this original equation into different forms that maintain its equivalence.

  1. The first option is \(y = -4x + 3\). This is solving for \(y\) in terms of \(x\) and is indeed equivalent to the original equation.

  2. The second option is \(x + 4y = 3\). To check if this is equivalent, we can manipulate it:

    • Start with \(x + 4y = 3\).
    • Rearranging gives \(4y = 3 - x\), or \(y = \frac{3 - x}{4}\).
    • To compare, we can try to express \(4x + y = 3\) in a similar form, or substitute this \(y\) back into \(4x + y = 3\) and see if we get a contradiction or equality.

When we multiply both sides of \(4x + y = 3\) by something to compare, we can find out equivalency.

Let’s try the other options:

  1. For the third option, \(8x + 2y = 6\):

    • Divide the entire equation by 2 gives \(4x + y = 3\), which is equivalent.

  2. The last option is \(3x + y = 4\). To check equivalency:

    • Rearranging gives \(y = 4 - 3x\). There’s no way from \(4x + y = 3\) to convert directly into \(3x + y = 4\), so they are not equivalent.

Thus, the two equations equivalent to \(4x + y = 3\) are:

  • \(y = -4x + 3\)
  • \(8x + 2y = 6\)