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How do you find an unknown value in an equation?

Published in Algebra Basics 3 mins read

You find an unknown value in an equation by isolating the variable that represents the unknown value on one side of the equation. This is achieved by using inverse operations.

Here's a breakdown of the process:

Understanding the Basics

  • Variable: The unknown value you're trying to find, typically represented by a letter (e.g., x, y, z).

  • Equation: A mathematical statement that shows two expressions are equal.

  • Inverse Operations: Operations that "undo" each other.

    • Addition and Subtraction are inverse operations.
    • Multiplication and Division are inverse operations.

Steps to Isolate the Variable

  1. Identify the Variable: Locate the variable you need to solve for.

  2. Apply Inverse Operations: Use inverse operations to eliminate any numbers or terms that are on the same side of the equation as the variable. Whatever operation you perform on one side of the equation, you must perform the same operation on the other side to maintain the equality.

  3. Simplify: After each step, simplify both sides of the equation.

  4. Repeat: Continue applying inverse operations and simplifying until the variable is isolated on one side of the equation.

Examples

Example 1: Solving for x in x + 5 = 10

  1. Variable: x
  2. Operation affecting x: + 5
  3. Inverse Operation: Subtraction
  4. Apply Inverse: Subtract 5 from both sides:
    • x + 5 - 5 = 10 - 5
  5. Simplify:
    • x = 5

Example 2: Solving for y in 2y = 6

  1. Variable: y
  2. Operation affecting y: 2 * y (multiplication)
  3. Inverse Operation: Division
  4. Apply Inverse: Divide both sides by 2:
    • (2y) / 2 = 6 / 2
  5. Simplify:
    • y = 3

Example 3: Solving for z in z/3 - 1 = 4

  1. Variable: z
  2. Operations affecting z: Division by 3 and subtraction by 1. We undo these in reverse order (opposite of PEMDAS/BODMAS).
  3. Inverse Operations: Addition and Multiplication.
  4. Apply Inverse (Addition): Add 1 to both sides:
    • z/3 - 1 + 1 = 4 + 1
    • z/3 = 5
  5. Apply Inverse (Multiplication): Multiply both sides by 3:
    • (z/3) 3 = 5 3
  6. Simplify:
    • z = 15

Key Considerations

  • Order of Operations: Remember to reverse the order of operations (PEMDAS/BODMAS) when applying inverse operations.
  • Maintaining Balance: Always perform the same operation on both sides of the equation to maintain equality.
  • Complex Equations: More complex equations may require multiple steps involving a combination of inverse operations and simplification.

By consistently applying these principles, you can effectively solve for unknown values in a wide range of equations.

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