If ` x cos 60^(@) + y cos 0^(@) = 3 and 4x sin 30^(@) - y cot 45^(@) = 2 ` , then what is the value of x ?

Options

A. `-1`
B. 0
C. 1
D. 2

Answer

To solve the equations given in the problem, we will follow these steps:

### Step 1: Write down the equations
We have two equations:
1. \( x \cos 60^\circ + y \cos 0^\circ = 3 \)
2. \( 4x \sin 30^\circ - y \cot 45^\circ = 2 \)

### Step 2: Substitute the trigonometric values
Using the known values of trigonometric functions:
- \( \cos 60^\circ = \frac{1}{2} \)
- \( \cos 0^\circ = 1 \)
- \( \sin 30^\circ = \frac{1}{2} \)
- \( \cot 45^\circ = 1 \)

Substituting these values into the equations, we get:
1. \( x \cdot \frac{1}{2} + y \cdot 1 = 3 \)
2. \( 4x \cdot \frac{1}{2} - y \cdot 1 = 2 \)

This simplifies to:
1. \( \frac{x}{2} + y = 3 \) (Equation 1)
2. \( 2x - y = 2 \) (Equation 2)

### Step 3: Rearranging Equation 1
From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 3 - \frac{x}{2} \]

### Step 4: Substitute \( y \) in Equation 2
Now, substitute \( y \) in Equation 2:
\[ 2x - (3 - \frac{x}{2}) = 2 \]

### Step 5: Simplify the equation
Expanding and simplifying:
\[ 2x - 3 + \frac{x}{2} = 2 \]
Multiplying the entire equation by 2 to eliminate the fraction:
\[ 4x - 6 + x = 4 \]
Combining like terms:
\[ 5x - 6 = 4 \]

### Step 6: Solve for \( x \)
Adding 6 to both sides:
\[ 5x = 10 \]
Dividing by 5:
\[ x = 2 \]

### Final Answer
The value of \( x \) is \( 2 \).

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