How To Solve 1 Step Equations

Mastering the Foundation: A Complete Guide to Solving One-Step Equations

Imagine you are standing on a perfectly balanced scale. On one side, you have a mysterious, sealed box representing an unknown value—let’s call it x. On the other side, you have a known weight of 10 pounds. The scale is level. Your goal is to figure out what’s inside the box without opening it. You realize that if you add 3 pounds to the side with the box, you must also add 3 pounds to the other side to keep it balanced. After doing so, the scale tips. You then remove 3 pounds from both sides to restore balance. Now, the box sits alone on one side, and a 7-pound weight sits on the other. You’ve solved the mystery: the box weighs 7 pounds. This intuitive process of maintaining balance is the exact heart of solving one-step equations. In its simplest form, a one-step equation is a mathematical statement where a variable (like x) is combined with a number using a single operation—addition, subtraction, multiplication, or division. Solving it means finding the value of that variable that makes the equation true, which we achieve by performing the inverse (opposite) operation on both sides of the equals sign. This fundamental skill is the critical first rung on the ladder of algebra, forming the bedrock for all future problem-solving in mathematics and science.

The Core Principle: The Law of Balance

Before diving into procedures, we must internalize the golden rule of solving equations: Whatever you do to one side of an equation, you must do to the other. An equation is a statement of equality, a declaration that two expressions represent the same value. The equals sign (=) is not a command to calculate; it is a symbol of perfect balance, like the beam of a scale. If you imagine the equation x + 5 = 12, you are saying that the value of x plus 5 is identical to 12. To isolate x, we need to "undo" the addition of 5. The inverse operation of addition is subtraction. Therefore, we subtract 5 from the left side. But to obey the law of balance, we must also subtract 5 from the right side. This yields x = 12 - 5, and finally x = 7. We can verify our solution by substituting 7 back into the original equation: 7 + 5 = 12, which is a true statement. This verification step is non-negotiable and builds confidence in your answer. The four inverse operation pairs are:

• Addition ↔ Subtraction
• Subtraction ↔ Addition
• Multiplication ↔ Division
• Division ↔ Multiplication

Understanding this principle transforms equation solving from a set of arbitrary rules into a logical, justifiable process. It’s about asking, "What operation is attached to the variable?" and then applying its opposite to both sides to free the variable.

Step-by-Step Breakdown by Operation Type

Let’s systematically walk through each of the four types of one-step equations, applying the inverse operation with precision.

1. Solving Equations with Addition (x + a = b)

• Goal: Isolate x.
• Operation: The variable is being added to a number (a). Use subtraction.
• Process: Subtract a from both sides.
• Example: Solve x + 8 = 15.
  • Subtract 8 from both sides: x + 8 - 8 = 15 - 8.
  • Simplify: x = 7.
  • Check: 7 + 8 = 15. ✔️ True.

2. Solving Equations with Subtraction (x - a = b)

• Goal: Isolate x.
• Operation: The variable is being subtracted by a number (a). Use addition.
• Process: Add a to both sides.
• Example: Solve x - 4 = 9.
  • Add 4 to both sides: x - 4 + 4 = 9 + 4.
  • Simplify: x = 13.
  • Check: 13 - 4 = 9. ✔️ True.

3. Solving Equations with Multiplication (a * x = b or ax = b)

• Goal: Isolate x.
• Operation: The variable is being multiplied by a number (a). Use division.
• Process: Divide both sides by a.
• Example: Solve 5x = 30.
  • Divide both sides by 5: (5x)/5 = 30/5.
  • Simplify: x = 6.
  • Check: 5 * 6 = 30. ✔️ True.
  • Note: If the coefficient is a fraction, like (1/3)x = 4, you would multiply both sides by 3 (the reciprocal) to isolate x.

4. Solving Equations with Division (x / a = b)

• Goal: Isolate x.
• Operation: The variable is being divided by a number (a). Use multiplication.
• Process: Multiply both sides by a.
• Example: Solve x / 2 = 10. *