Understanding Równania Wymierne (Rational Equations)

This page provides a detailed overview of równania wymierne (rational equations) and methods for solving them. It covers essential concepts and techniques that students need to master when dealing with these types of mathematical problems.

Definition: Równania wymierne are equations that contain fractions where the numerator, denominator, or both contain variables.

The page outlines the general approach to solving rational equations:

1. Simplify the equation by bringing all terms to one side, setting it equal to zero.
2. Find a common denominator for all fractions in the equation.
3. Multiply both sides of the equation by the common denominator to eliminate fractions.
4. Solve the resulting polynomial equation.

Highlight: Always remember to check the domain of the equation to avoid division by zero!

Several examples are provided to illustrate these steps:

Example: Solving 6/(x+2) - 1 = 0

1. Multiply both sides by (x+2): 6 - (x+2) = 0
2. Simplify: 6 - x - 2 = 0
3. Solve: -x = -4, so x = 4

Example: Solving (x+1)/(x-2) = x/(x+3)

1. Cross-multiply: (x+1)(x+3) = x(x-2)
2. Expand: x² + 4x + 3 = x² - 2x
3. Simplify: 6x + 3 = 0
4. Solve: x = -1/2

The page also covers more complex examples, including equations with quadratic terms and those requiring factoring.

Vocabulary: Dziedzina (Domain) - The set of all possible values for the variable that make the equation valid.

The importance of checking the domain is emphasized throughout the examples, ensuring students understand that solutions must be verified within the context of the original equation.