Find the value of x and y I. `18x^(2)-9x+1=0`
II. `12y^(2)-y-1=0`

Options

A. (a) x = 1/3 , 1/6
y = 1/3 , -1/4
B. (b) x = 5 , -3
y = -1 , 2
C. (c) x = 5/2 , 2/3
y = -1/3 , -1/4
D. (d) x = 1 , 0
y = 5 , 2

Answer

To solve the equations provided, we will go through each equation step by step.

### Step 1: Solve for x in the equation \( 18x^2 - 9x + 1 = 0 \)

1. **Identify the coefficients**:
- \( a = 18 \)
- \( b = -9 \)
- \( c = 1 \)

2. **Use the quadratic formula**:
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

3. **Calculate the discriminant**:
\[ b^2 - 4ac = (-9)^2 - 4 \cdot 18 \cdot 1 = 81 - 72 = 9 \]

4. **Substitute values into the quadratic formula**:
\[ x = \frac{-(-9) \pm \sqrt{9}}{2 \cdot 18} = \frac{9 \pm 3}{36} \]

5. **Calculate the two possible values for x**:
- First value:
\[ x_1 = \frac{9 + 3}{36} = \frac{12}{36} = \frac{1}{3} \]
- Second value:
\[ x_2 = \frac{9 - 3}{36} = \frac{6}{36} = \frac{1}{6} \]

### Step 2: Solve for y in the equation \( 12y^2 - y - 1 = 0 \)

1. **Identify the coefficients**:
- \( a = 12 \)
- \( b = -1 \)
- \( c = -1 \)

2. **Use the quadratic formula**:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

3. **Calculate the discriminant**:
\[ b^2 - 4ac = (-1)^2 - 4 \cdot 12 \cdot (-1) = 1 + 48 = 49 \]

4. **Substitute values into the quadratic formula**:
\[ y = \frac{-(-1) \pm \sqrt{49}}{2 \cdot 12} = \frac{1 \pm 7}{24} \]

5. **Calculate the two possible values for y**:
- First value:
\[ y_1 = \frac{1 + 7}{24} = \frac{8}{24} = \frac{1}{3} \]
- Second value:
\[ y_2 = \frac{1 - 7}{24} = \frac{-6}{24} = -\frac{1}{4} \]

### Final Results
The solutions are:
- For \( x \): \( \frac{1}{3} \) and \( \frac{1}{6} \)
- For \( y \): \( \frac{1}{3} \) and \( -\frac{1}{4} \)