Solving Linear Equations Formula

The Formula

For ax + b = c: x = \frac{c - b}{a}, \quad a \neq 0

When to use: Undo what's done to x by doing the opposite: if x + 5, subtract 5.

Quick Example

2x + 3 = 11 — subtract 3 to get 2x = 8, then divide by 2 to get x = 4.

Notation

ax + b = c where a is the coefficient, b is the constant term, and x is the variable to isolate.

What This Formula Means

The process of finding the value of the variable that makes a linear equation true, using inverse operations to isolate the variable on one side of the equals sign. A linear equation has the variable raised only to the first power, producing exactly one solution.

Undo what's done to x by doing the opposite: if x + 5, subtract 5.

Formal View

\forall a, b, c \in \mathbb{R},\; a \neq 0: \; ax + b = c \iff x = \frac{c - b}{a} (unique solution in \mathbb{R}).

Worked Examples

Example 1

easy
Solve 3x + 7 = 22.

Solution

  1. 1
    Subtract 7 from both sides: 3x = 22 - 7 = 15.
  2. 2
    Divide both sides by 3: x = \frac{15}{3} = 5.
  3. 3
    Check: 3(5) + 7 = 15 + 7 = 22 ✓

Answer

x = 5
To solve a linear equation, isolate x by performing inverse operations. Always verify your answer by substituting back into the original equation.

Example 2

medium
Solve 2(x - 3) + 4 = 3x - 8.

Example 3

medium
Solve 2(3x - 4) = 5x + 6.

Common Mistakes

  • Forgetting to perform the same operation on both sides of the equation
  • Making sign errors when moving terms across the equals sign — subtracting instead of adding
  • Not simplifying both sides before isolating the variable, leading to unnecessary complexity

Why This Formula Matters

Solving linear equations is the core skill underlying all higher mathematics and practical real-world problem solving.

Frequently Asked Questions

What is the Solving Linear Equations formula?

The process of finding the value of the variable that makes a linear equation true, using inverse operations to isolate the variable on one side of the equals sign. A linear equation has the variable raised only to the first power, producing exactly one solution.

How do you use the Solving Linear Equations formula?

Undo what's done to x by doing the opposite: if x + 5, subtract 5.

What do the symbols mean in the Solving Linear Equations formula?

ax + b = c where a is the coefficient, b is the constant term, and x is the variable to isolate.

Why is the Solving Linear Equations formula important in Math?

Solving linear equations is the core skill underlying all higher mathematics and practical real-world problem solving.

What do students get wrong about Solving Linear Equations?

Order matters—undo addition/subtraction before multiplication/division.

What should I learn before the Solving Linear Equations formula?

Before studying the Solving Linear Equations formula, you should understand: equations, order of operations.

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