Solving Linear Equations: Concepts, Methods, and Classification

Solving Linear Equations

Linear Expressions vs. Linear Equations

A linear expression is an algebraic expression involving a variable raised only to the first power, such as $2x + 3$. A linear equation sets a linear expression equal to a value, such as $2x + 3 = 5$.

• Linear Expression: $2x + 3$

• Linear Equation: $2x + 3 = 5$

To solve a linear equation, find the value(s) of the variable that make the equation true.

Solving Linear Equations: Basic Steps

• Isolate the variable (usually $x$) using inverse operations (addition, subtraction, multiplication, division).

• Always perform the same operation on both sides of the equation to maintain equality.

• Multiple operations may be needed to fully isolate the variable.

Example:

Solve $2x + 3 = 5$

• Subtract 3 from both sides: $2x = 2$

• Divide both sides by 2: $x = 1$

Key Operations for Solving Linear Equations

• Addition/Subtraction: Used to move constants or variables from one side to the other.

• Multiplication/Division: Used to eliminate coefficients of the variable.

Example:

Solve $x + 2 = 0$

• Subtract 2 from both sides: $x = -2$

General Steps for Solving Linear Equations

1. Distribute constants (if necessary)

2. Combine like terms

3. Group terms with $x$ and constants on opposite sides

4. Isolate $x$ (solve for $x$)

5. Check the solution by substituting back into the original equation

Example:

Solve $2(x - 3) = 0$

• Distribute: $2x - 6 = 0$

• Add 6 to both sides: $2x = 6$

• Divide by 2: $x = 3$

Solving Linear Equations with Fractions

Linear equations may contain fractions. To simplify, eliminate fractions by multiplying both sides by the Least Common Denominator (LCD).

Example:

Solve $\frac{1}{2}(x + 2) - \frac{1}{3}x = \frac{1}{2}$

• Multiply both sides by 6 (LCD): $3(x + 2) - 2x = 3$

• Expand: $3x + 6 - 2x = 3$

• Combine like terms: $x + 6 = 3$

• Subtract 6: $x = -3$

Step-by-Step Table: Solving Linear Equations

Step	Description
1	Multiply by LCD to eliminate fractions (if needed)
2	Distribute constants
3	Combine like terms
4	Group terms with $x$ and constants on opposite sides
5	Isolate $x$ (if any $x$ terms remain)
6	Check solution by substituting in original equation (optional)

Categorizing Linear Equations

Types of Linear Equations Based on Solutions

Linear equations can be classified by the number of solutions they have:

• Conditional: Has exactly one solution.

• Identity: True for all real numbers (infinite solutions).

• Inconsistent: No solution (contradiction).

Table: Classification of Linear Equations

Type	Example	Solution	Explanation
Conditional	$2x + 4 = 10$	$x = 3$	One solution
Identity	$x + 5 = x + 5$	All real numbers	True for any $x$
Inconsistent	$x + 4 = x + 2$	No solution	Contradiction (false statement)

Example:

• Solve $2x + 4 = 10$

• $2x = 6 \implies x = 3$ (Conditional)

• Solve $x + 5 = x + 5$

• True for all $x$ (Identity)

• Solve $x + 4 = x + 2$

• $4 = 2$ (False, so Inconsistent)

Practice Problems

• Solve $3(2 - 5x) = 4x + 25$

• Solve $\frac{1}{2} + (x + 2) = \frac{3}{4}x - 1$

• Solve and classify: $5x + 17 = 8x + 12 - 3(x + 4)$

• Solve and classify: $\frac{x}{4} + \frac{1}{6} = \frac{x}{3} + 4$

• Solve and classify: $-2(5 - 3x) + x = 7x - 10$

Summary

• Linear equations are solved by isolating the variable using inverse operations.

• Always perform the same operation on both sides of the equation.

• Equations can be classified as conditional, identity, or inconsistent based on their solutions.

• Fractions can be cleared using the LCD before solving.