Solve each of the following equatins :
`5x-(35)/(x)=18,xne0`

Answer

To solve the equation \( 5x - \frac{35}{x} = 18 \) with the condition \( x \neq 0 \), we will follow these steps:

### Step 1: Eliminate the fraction
Multiply both sides of the equation by \( x \) (since \( x \neq 0 \)):
\[ x(5x) - 35 = 18x \]
This simplifies to:
\[ 5x^2 - 35 = 18x \]

### Step 2: Rearrange the equation
Rearranging the equation gives us:
\[ 5x^2 - 18x - 35 = 0 \]

### Step 3: Factor the quadratic equation
Now we need to factor the quadratic equation \( 5x^2 - 18x - 35 = 0 \). To do this, we look for two numbers that multiply to \( 5 \times -35 = -175 \) and add up to \( -18 \). The numbers that fit this are \( -25 \) and \( 7 \).

### Step 4: Rewrite the middle term
We can rewrite the equation as:
\[ 5x^2 - 25x + 7x - 35 = 0 \]

### Step 5: Factor by grouping
Now we can group the terms:
\[ (5x^2 - 25x) + (7x - 35) = 0 \]
Factoring out the common factors in each group:
\[ 5x(x - 5) + 7(x - 5) = 0 \]

### Step 6: Factor out the common binomial
Now we can factor out \( (x - 5) \):
\[ (5x + 7)(x - 5) = 0 \]

### Step 7: Solve for x
Setting each factor to zero gives us:
1. \( 5x + 7 = 0 \)
\[ 5x = -7 \implies x = -\frac{7}{5} \]
2. \( x - 5 = 0 \)
\[ x = 5 \]

### Step 8: Final solutions
Thus, the solutions to the equation are:
\[ x = 5 \quad \text{and} \quad x = -\frac{7}{5} \]