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How to Isolate The Variable in an Equation

Numbers are easier to manipulate than things. If you're a cattle rancher, for example, it's easier to subtract 1,000 head of cattle from 1,500 head of cattle and know that leaves 500 cattle (1500 - 1000 = 500) than to go out on the range, herd and pen 1,500 cattle, separate 1,000 from the 1,500, and finally count the remainder. The manipulative characteristics of numbers are especially useful when you're simplifying equations, or isolating the variable in an equation. Isolating the variable in an equation is often your first and most important step when solving an equation.

Instructions

Method One -- Addition or Subtraction
- 1
  Write the equation you want to analyze. For example, x - 7 = 6.
- 2
  Identify a mathematical operation that when performed on both sides of the equal sign (=) will leave the variable, x, alone on its side of the equal sign. For example, note that adding 7 on both sides of the equal sign will leave x alone on the left side of the equation.
- 3
  Write the operation you have identified. For example, x - 7 + 7 = 6 + 7.
- 4
  Perform the operations you have written. For example, x = 13.
Method Two -- Multiplication
- 5
  Write the equation you want to analyze. For example, 3x = 21.
- 6
  Remember that by definition, the product of a number and its inverse is 1. The inverse of a number is 1 divided by that number. For example, the inverse of 3 is 1/3, and 3 * (1/3) = 1.
- 7
  Multiply both sides of your equation by the inverse of the coefficient of the variable in your equation. The coefficient of a variable is the number immediately preceding the variable. For example, the coefficient of 3x is 3. Write the equation. (1/3) * 3x = (1/3) * 21.
- 8
  Perform the operations you have written. For example, x = 7.

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