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

Using inverse operations in reverse order to isolate the variable and find its exact numerical value.

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

  • Wrong order of operations when solving
  • Sign errors with negatives

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?

Using inverse operations in reverse order to isolate the variable and find its exact numerical value.

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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