Solve each of the following equatins :
`x=(3x+1)/(4x)`

Answer

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

### Step 1: Eliminate the fraction
Multiply both sides of the equation by \( 4x \) to eliminate the denominator.

\[ 4x \cdot x = 3x + 1 \]

This simplifies to:

\[ 4x^2 = 3x + 1 \]

### Step 2: Rearrange the equation
Move all terms to one side of the equation to set it to zero.

\[ 4x^2 - 3x - 1 = 0 \]

### Step 3: Factor the quadratic equation
Now we need to factor the quadratic equation \( 4x^2 - 3x - 1 \). We look for two numbers that multiply to \( 4 \times (-1) = -4 \) and add to \( -3 \). The numbers \( -4 \) and \( 1 \) work.

Rewriting the middle term:

\[ 4x^2 - 4x + x - 1 = 0 \]

### Step 4: Group the terms
Group the terms:

\[ (4x^2 - 4x) + (x - 1) = 0 \]

### Step 5: Factor by grouping
Factor out the common terms from each group:

\[ 4x(x - 1) + 1(x - 1) = 0 \]

Now factor out \( (x - 1) \):

\[ (x - 1)(4x + 1) = 0 \]

### Step 6: Solve for \( x \)
Set each factor to zero:

1. \( x - 1 = 0 \) leads to \( x = 1 \)
2. \( 4x + 1 = 0 \) leads to \( 4x = -1 \) or \( x = -\frac{1}{4} \)

### Final Solutions
The solutions to the equation are:

\[ x = 1 \quad \text{and} \quad x = -\frac{1}{4} \]

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