Solving Linear Equations with Fractions

Solving Linear Equations

Linear Equations with Fractions

Linear equations are equations of the first degree, meaning the variable appears only to the first power. When fractions are involved, the process of solving remains systematic: clear denominators, isolate the variable, and solve.

• Definition: A linear equation in one variable is an equation that can be written in the form $ax + b = c$, where $a$, $b$, and $c$ are constants.

• Key Steps:

  1. Clear fractions by multiplying both sides by the least common denominator (LCD).

  2. Combine like terms and isolate the variable.

  3. Solve for the variable.

  4. Check the solution in the original equation.

Example Problem

Given Equation:

$\dfrac{x + 6}{2} = 2 - \dfrac{x + 6}{3}$

• Step 1: Clear Fractions Multiply both sides by 6 (the LCD of 2 and 3): $6 \left( \dfrac{x + 6}{2} \right) = 6 \left( 2 - \dfrac{x + 6}{3} \right)$ $3(x + 6) = 12 - 2(x + 6)$

• Step 2: Expand and Combine Like Terms $3x + 18 = 12 - 2x - 12$ $3x + 18 = -2x$ Add $2x$ to both sides: $3x + 2x + 18 = 0$ $5x + 18 = 0$

• Step 3: Isolate the Variable $5x = -18$ $x = -\dfrac{18}{5}$

• Step 4: Solution Set The solution set is $\left\{ -\dfrac{18}{5} \right\}$.

Types of Solution Sets

• Single Solution: The equation has one unique solution (as in this example).

• All Real Numbers: The equation is true for all real numbers (an identity).

• No Solution: The equation is never true (contradiction).

Summary Table: Solution Set Types

Type	Description	Example
Single Solution	One value of $x$ satisfies the equation	$x = -\dfrac{18}{5}$
All Real Numbers	Any real number is a solution	$x + 1 = x + 1$
No Solution	No value of $x$ satisfies the equation	$x + 1 = x + 2$

Application

• Solving equations with fractions is a foundational skill for algebra and precalculus, useful in modeling real-world problems involving rates, proportions, and more.