Solving Linear Equations with Fractions: Step-by-Step Guide

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:

$\frac{9t}{20} - \frac{t}{5} = 1$

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

2. Multiply both sides by 20: $20 \left( \frac{9t}{20} - \frac{t}{5} \right ) = 20 \cdot 1$ $20 \left( \frac{9t}{20} \right ) - 20 \left( \frac{t}{5} \right ) = 20$ $9t - 4t = 20$

3. Combine like terms: $5t = 20$

4. Isolate t: $\frac{5t}{5} = \frac{20}{5}$ $t = 4$

5. Check the solution: Substitute $t = 4$ into the original equation: $\frac{9 \cdot 4}{20} - \frac{4}{5} = 1$ $\frac{36}{20} - \frac{16}{20} = 1$ $\frac{20}{20} = 1$ $1 = 1$ Since a true statement results, $t = 4$ is the solution.

Solution Set: $\{4\}$

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