If `x^(12)=(3)/(z) and x^(11)=(3y)/(z)` which of the following is an expression for x in terms of y?

Options

A. 3y
B. 2y
C. y
D. `(1)/(y)`

Answer

To solve the problem, we start with the given equations:

1. \( x^{12} = \frac{3}{z} \) (Equation 1)
2. \( x^{11} = \frac{3y}{z} \) (Equation 2)

### Step 1: Express \( z \) in terms of \( y \) and \( x \)

From Equation 2, we can isolate \( z \):

\[ z = \frac{3y}{x^{11}} \]

### Step 2: Substitute \( z \) into Equation 1

Now, we substitute this expression for \( z \) back into Equation 1:

\[ x^{12} = \frac{3}{\left(\frac{3y}{x^{11}}\right)} \]

### Step 3: Simplify the right-hand side

When we simplify the right-hand side, we get:

\[ x^{12} = \frac{3 \cdot x^{11}}{3y} \]

The 3s cancel out:

\[ x^{12} = \frac{x^{11}}{y} \]

### Step 4: Rearranging the equation

Now, we can rearrange the equation to isolate \( x \):

\[ x^{12} \cdot y = x^{11} \]

### Step 5: Divide both sides by \( x^{11} \)

Dividing both sides by \( x^{11} \) (assuming \( x \neq 0 \)) gives:

\[ y \cdot x = 1 \]

### Step 6: Solve for \( x \)

Finally, we solve for \( x \):

\[ x = \frac{1}{y} \]

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

\[ x = \frac{1}{y} \]