Linear Equation Calculator

A linear equation is a first-degree polynomial equation in one variable, taking the general form:

$ax + b = 0$

where $a$ and $b$ are constants, and $a \neq 0$. The word "linear" comes from the fact that the graph of such an equation is a straight line.

More generally, a linear equation in one variable can appear as:

$ax + b = cx + d$

which can always be rearranged into the standard form. The solution is the value of $x$ that makes both sides of the equation equal.

Linear equations are the foundation of algebra and appear everywhere in real life — from calculating costs and distances to converting units and balancing budgets. They always have exactly one solution (assuming $a \neq 0$), which makes them the simplest type of equation to solve.

Key characteristics of linear equations:

• The variable $x$ appears only to the first power (no $x^2$, $\sqrt{x}$, etc.)
• The graph is always a straight line
• There is exactly one solution
• They can always be solved in a finite number of algebraic steps

Solving a linear equation means isolating the variable on one side. Here are the main approaches:

1. Basic Isolation Method

For equations in the form $ax + b = c$:

1. Subtract $b$ from both sides: $ax = c - b$
2. Divide both sides by $a$: $x = \frac{c - b}{a}$

Example: Solve $3x + 7 = 22$

• $3x = 22 - 7 = 15$
• $x = \frac{15}{3} = 5$

2. Variables on Both Sides

For equations like $ax + b = cx + d$:

1. Move all variable terms to one side: $(a - c)x + b = d$
2. Move constants to the other side: $(a - c)x = d - b$
3. Divide: $x = \frac{d - b}{a - c}$

Example: Solve $4x - 3 = 2x + 9$

• $4x - 2x = 9 + 3$
• $2x = 12$
• $x = 6$

3. Equations with Parentheses

First distribute, then collect like terms:

Example: Solve $5(x - 2) = 3x + 4$

• $5x - 10 = 3x + 4$
• $2x = 14$
• $x = 7$

4. Equations with Fractions

Multiply both sides by the LCD to eliminate fractions:

Example: Solve $\frac{x}{3} + 2 = 7$

• Multiply by 3: $x + 6 = 21$
• $x = 15$

• Forgetting to apply operations to both sides: Whatever you do to one side, you must do to the other.
• Sign errors when moving terms: When moving $+5$ to the other side, it becomes $-5$, not $+5$.
• Not distributing correctly: $3(x - 4) = 3x - 12$, not $3x - 4$.
• Dividing by zero: If you end up with $0x = 5$, the equation has no solution; if $0x = 0$, it has infinitely many solutions.
• Forgetting to simplify fractions: Always reduce your final answer to lowest terms.