Solve each of the following equatins :
`(x+1)/(x-1)=(3x-7)/(2x-3)`

Answer

To solve the equation \((x+1)/(x-1)=(3x-7)/(2x-3)\), we will follow these steps:

### Step 1: Cross-Multiply
We start by cross-multiplying the fractions to eliminate the denominators:

\[ (x + 1)(2x - 3) = (3x - 7)(x - 1) \]

### Step 2: Expand Both Sides
Now we will expand both sides of the equation:

Left Side:
\[ (x + 1)(2x - 3) = 2x^2 - 3x + 2x - 3 = 2x^2 - x - 3 \]

Right Side:
\[ (3x - 7)(x - 1) = 3x^2 - 3x - 7x + 7 = 3x^2 - 10x + 7 \]

So, we have:
\[ 2x^2 - x - 3 = 3x^2 - 10x + 7 \]

### Step 3: Move All Terms to One Side
Next, we will move all terms to one side of the equation to set it to zero:

\[ 2x^2 - x - 3 - 3x^2 + 10x - 7 = 0 \]

Combine like terms:
\[ - x^2 + 9x - 10 = 0 \]

### Step 4: Rearrange the Equation
We can multiply through by -1 to make the leading coefficient positive:

\[ x^2 - 9x + 10 = 0 \]

### Step 5: Factor the Quadratic Equation
Now we will factor the quadratic equation:

\[ x^2 - 9x + 10 = (x - 10)(x + 1) = 0 \]

### Step 6: Solve for x
Set each factor equal to zero:

1. \(x - 10 = 0 \Rightarrow x = 10\)
2. \(x + 1 = 0 \Rightarrow x = -1\)

### Final Solution
The solutions to the equation are:
\[ x = 10 \quad \text{and} \quad x = -1 \]

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