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HH.3 Evaluate variable equations

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What are variable equations?

Variable equations are mathematical statements that contain an equals sign and at least one variable. A variable is a symbol, usually a letter, that represents an unknown value.

Examples:
  • n + 7 = 15
  • 3 × y = 24
  • 45 ÷ x = 9
Note

Think of a variable as a mystery number waiting to be discovered. Your job is to find which number makes the equation true.

How to evaluate variable equations

To evaluate a variable equation means to find the value of the variable that makes the equation true. You do this by performing inverse operations to isolate the variable.

Steps:
  • Identify the operation being performed on the variable
  • Perform the opposite (inverse) operation on both sides
  • Simplify both sides of the equation
  • Check your solution by substituting it back into the original equation
Note

Always perform the same operation on both sides of the equals sign to keep the equation balanced, just like a scale.

Solving addition and subtraction equations

When a variable has a number added to it, subtract that number from both sides. When a variable has a number subtracted from it, add that number to both sides.

Examples:
  • x + 12 = 30 → x = 30 - 12 → x = 18
  • y - 8 = 15 → y = 15 + 8 → y = 23
  • 17 = a + 9 → 17 - 9 = a → 8 = a
Note

Addition and subtraction are inverse operations - they undo each other.

Solving multiplication and division equations

When a variable is multiplied by a number, divide both sides by that number. When a variable is divided by a number, multiply both sides by that number.

Examples:
  • 6m = 42 → m = 42 ÷ 6 → m = 7
  • n ÷ 5 = 9 → n = 9 × 5 → n = 45
  • 28 = 4k → 28 ÷ 4 = k → 7 = k
Note

Multiplication and division are inverse operations - they undo each other.

Equations with multiple operations

Some equations require two steps to solve. First, undo addition or subtraction, then undo multiplication or division.

Examples:
  • 3x + 5 = 20 → 3x = 15 → x = 5
  • y ÷ 4 - 2 = 3 → y ÷ 4 = 5 → y = 20
  • 2a - 7 = 11 → 2a = 18 → a = 9
Note

Think of the equation like getting dressed: the last operation put on is the first one you take off when solving.

Using equations with two variables

Some equations contain two variables. When you know the value of one variable, you can substitute it into the equation to find the value of the other variable.

Example:
  • Use the equation h = k + 5 to find the value of h when k = 5
  • Substitute 5 for k: h = 5 + 5
  • Simplify: h = 10
Another Example:
  • Use the equation m = 3 × n to find the value of m when n = 7
  • Substitute 7 for n: m = 3 × 7
  • Simplify: m = 21
Note

When substituting values, replace the variable with the given number and perform the calculation carefully. Always double-check your arithmetic.

Checking your solutions

Always verify your answer by substituting it back into the original equation to ensure both sides are equal.

Example:
  • Equation: x + 8 = 17
  • Solution: x = 9
  • Check: 9 + 8 = 17 ✓
Note

Checking your work helps catch calculation errors and confirms your solution is correct.