Solving Linear Equations
A linear equation is any equation where the variable appears only to the first power. Solving it means finding the value of the variable that makes both sides equal.
The Golden Rule of Equations
ax + b = c → x = (c − b) / a
Whatever operation you apply to one side, apply it identically to the other side.
Concept Explanation
A linear equation looks like ax + b = c, where a, b, and c are numbers and x is the unknown you need to find. The goal is to get x by itself on one side of the equals sign.
The key principle is the Balance Rule: whatever you do to one side of the equation, you must do exactly the same to the other side. This keeps the equation true throughout every step.
You solve by performing inverse operations — if a number is added to x, subtract it from both sides; if x is multiplied by a number, divide both sides by that number. Work from the outside in: handle addition/subtraction first, then multiplication/division.
Always check your answer by substituting it back into the original equation. If both sides evaluate to the same number, your solution is correct.
Worked Examples
Write the equation
3x + 7 = 22
Subtract 7 from both sides
3x + 7 − 7 = 22 − 7 → 3x = 15
Divide both sides by 3
3x ÷ 3 = 15 ÷ 3 → x = 5
Check: substitute x = 5
3(5) + 7 = 15 + 7 = 22 ✓
Answer: x = 5
Key Terms
Variable
A letter (usually x or y) representing an unknown number.
Coefficient
The number multiplied by the variable. In 3x, the coefficient is 3.
Inverse Operation
The opposite operation used to undo an action. Addition ↔ Subtraction; Multiplication ↔ Division.
Solution
The value of the variable that makes the equation true.
Balance Rule
Whatever you do to one side of an equation, you must do to the other.
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Solving Linear Equations