Solve each equation. x/(x-3) = 3/(x-3) + 3

Verified step by step guidance

1. Identify the given equation: \(\frac{x}{x-3} = \frac{3}{x-3} + 3\).
2. Notice that the denominators on the left and right side are the same, \(x-3\), so we should consider the domain restriction: \(x \neq 3\) to avoid division by zero.
3. Multiply both sides of the equation by the common denominator \((x-3)\) to eliminate the fractions: \(\left(\frac{x}{x-3}\right)(x-3) = \left(\frac{3}{x-3} + 3\right)(x-3)\).
4. Simplify both sides after multiplication: the left side becomes \(x\), and the right side becomes \$3 + 3(x-3)$.
5. Set up the resulting equation without fractions: \(x = 3 + 3(x - 3)\), then solve for \(x\) by expanding and isolating the variable.

Key Concepts

Solving Rational Equations

Rational equations involve fractions with polynomials in the numerator and denominator. To solve them, identify common denominators and eliminate fractions by multiplying both sides by the least common denominator (LCD), simplifying the equation to a polynomial form.

Restrictions on the Variable

When solving rational equations, values that make any denominator zero are excluded from the solution set. These restrictions must be identified before solving to avoid invalid solutions that cause division by zero.

Isolating and Solving Linear Equations

After clearing denominators, the resulting equation is often linear. Use algebraic techniques like combining like terms and isolating the variable to find the solution. Verify solutions against restrictions to ensure validity.