Equations, Inequalities, and Problem Solving

Solving Linear Equations with Fractions

Linear equations often contain fractional terms, which can make solving them more complex. A systematic approach involves clearing fractions by multiplying both sides by the least common denominator (LCD), simplifying, and then isolating the variable.

• Key Point 1: Clear fractions using the LCD. Multiply both sides of the equation by the LCD of all denominators to eliminate fractions.

• Key Point 2: Apply the distributive property. Distribute the LCD across all terms in the equation.

• Key Point 3: Combine like terms and isolate the variable. Simplify the resulting equation and use properties of equality to solve for the variable.

• Key Point 4: Check the solution. Substitute the found value back into the original equation to verify it produces a true statement.

Example: Solving a Fractional Linear Equation

Consider the equation:

1. Find the LCD: The denominators are 20 and 5. The LCD is 20.

2. Multiply both sides by 20:

3. Combine like terms:

4. Isolate t:

5. Check the solution: Substitute into the original equation: Since a true statement results, is the solution.

Solution Set:

Additional info: This method is applicable to any linear equation with fractional coefficients. Always check your solution by substituting back into the original equation.