In the formula `(3y)/(2x -y) - 2 = 5x`, what is the correct expression of y in terms of x ?

Options

A. `(2x(5x +2))/(5(x+1))`
B. `(x(5x+2))/((x+5))`
C. `(5x(5x+1))/(2(x+1))`
D. `(5x(x+5))/(2(5x+2))`

Answer

To solve the equation \(\frac{3y}{2x - y} - 2 = 5x\) for \(y\) in terms of \(x\), we will follow these steps:

### Step 1: Isolate the fraction
Start by adding \(2\) to both sides of the equation to isolate the fraction:

\[ \frac{3y}{2x - y} = 5x + 2 \]

**Hint:** To isolate a term, you can add or subtract the same value from both sides of the equation.

### Step 2: Eliminate the fraction
Multiply both sides of the equation by \(2x - y\) to eliminate the fraction:

\[ 3y = (5x + 2)(2x - y) \]

**Hint:** When multiplying both sides by a term that includes a variable, ensure that the term does not equal zero.

### Step 3: Expand the right side
Now, expand the right side of the equation:

\[ 3y = (5x)(2x) + (5x)(-y) + (2)(2x) + (2)(-y) \]
\[ 3y = 10x^2 - 5xy + 4x - 2y \]

**Hint:** Use the distributive property to expand expressions.

### Step 4: Move all terms involving \(y\) to one side
Rearrange the equation to get all terms involving \(y\) on one side:

\[ 3y + 5xy + 2y = 10x^2 + 4x \]
\[ (3y + 2y + 5xy) = 10x^2 + 4x \]
\[ (5y + 5xy) = 10x^2 + 4x \]

**Hint:** You can combine like terms to simplify your equation.

### Step 5: Factor out \(y\)
Factor \(y\) out from the left side:

\[ y(5 + 5x) = 10x^2 + 4x \]

**Hint:** Factoring can simplify the equation and make it easier to isolate the variable.

### Step 6: Solve for \(y\)
Now, divide both sides by \(5 + 5x\) to solve for \(y\):

\[ y = \frac{10x^2 + 4x}{5 + 5x} \]

**Hint:** To isolate a variable, you can divide both sides by the coefficient of that variable.

### Final Expression
Thus, the expression for \(y\) in terms of \(x\) is:

\[ y = \frac{10x^2 + 4x}{5 + 5x} \]