Solve: `2y+11/4=1/3y+2`.

Options

A. `y=-7/20`
B. `y=-5/20`
C. `y=-3/20`
D. `y=-9/20`

Answer

To solve the equation \(2y + \frac{11}{4} = \frac{1}{3}y + 2\), we will follow these steps:

### Step 1: Rearrange the equation
We want to get all terms involving \(y\) on one side and constant terms on the other side. We can do this by subtracting \(\frac{1}{3}y\) from both sides and subtracting \(\frac{11}{4}\) from both sides.

\[ 2y - \frac{1}{3}y = 2 - \frac{11}{4} \]

### Step 2: Find a common denominator
To simplify the left side, we need to find a common denominator for the \(y\) terms. The common denominator for \(1\) and \(3\) is \(3\).

\[ 2y = \frac{6}{3}y \]

So, we rewrite the left side:

\[ \frac{6}{3}y - \frac{1}{3}y = \frac{5}{3}y \]

Now, we rewrite the right side. The common denominator for \(2\) and \(\frac{11}{4}\) is \(4\).

\[ 2 = \frac{8}{4} \]

So, we rewrite the right side:

\[ \frac{8}{4} - \frac{11}{4} = \frac{8 - 11}{4} = \frac{-3}{4} \]

### Step 3: Set the equation
Now we have:

\[ \frac{5}{3}y = \frac{-3}{4} \]

### Step 4: Solve for \(y\)
To isolate \(y\), we multiply both sides by the reciprocal of \(\frac{5}{3}\), which is \(\frac{3}{5}\):

\[ y = \frac{-3}{4} \times \frac{3}{5} \]

### Step 5: Multiply the fractions
Now, we multiply the fractions:

\[ y = \frac{-3 \times 3}{4 \times 5} = \frac{-9}{20} \]

### Final Answer
Thus, the solution to the equation is:

\[ y = \frac{-9}{20} \]

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