Which of the following equations is equivalent to $$ y=\frac{2}{3} x+\frac{1}{4} $$ ? $$ y=8 x+3 $$ $$ 12 y=8 x+3 $$ $$ 12 y=2 x+1 $$

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To determine which of the given equations is equivalent to $$ y=\frac{2}{3} x+\frac{1}{4} $$, we can manipulate the original equation and compare it with the options provided.

1. Start with the original equation:
   $$
   y = \frac{2}{3} x + \frac{1}{4}
   $$

2. To eliminate the fractions, we can multiply both sides of the equation by a common denominator. The least common multiple of the denominators (3 and 4) is 12. So, we multiply the entire equation by 12:
   $$
   12y = 12 \left( \frac{2}{3} x + \frac{1}{4} \right)
   $$

3. Distributing 12 on the right side:
   $$
   12y = 12 \cdot \frac{2}{3} x + 12 \cdot \frac{1}{4}
   $$

4. Simplifying each term:
   $$
   12 \cdot \frac{2}{3} = 8 \quad \text{and} \quad 12 \cdot \frac{1}{4} = 3
   $$
   Thus, we have:
   $$
   12y = 8x + 3
   $$

Now we can compare this result with the options provided:

- $$ y = 8x + 3 $$ (not equivalent)
- $$ 12y = 8x + 3 $$ (equivalent)
- $$ 12y = 2x + 1 $$ (not equivalent)

The equation that is equivalent to $$ y=\frac{2}{3} x+\frac{1}{4} $$ is:
$$
\boxed{12y = 8x + 3}
$$
The equivalent equation is $$ 12y = 8x + 3 $$.

Which of the following equations is equivalent to \( y=\frac{2}{3} x+\frac{1}{4} \) ? \( y=8 x+3 \) \( 12 y=8 x+3 \) \( 12 y=2 x+1 \)

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