One-step and two-step equations and inequalities

1. Structure of linear equations and unknowns

You’ll move much faster in algebra if you can look at a problem and see “what’s missing” and “what’s given.”

A linear equation is a math sentence with:

• an unknown (usually a letter like x or y)
• numbers (things you know)
• operations (add, subtract, multiply, divide)
• an equals sign =, which says both sides have the same value

What you’re solving for

The letter is called a variable.
Your goal is to find the value of the variable that makes the equation true.

Example:

• x + 5 = 12
  • Variable (unknown): x
  • Given numbers: 5 and 12
  • Meaning: “Some number plus 5 is 12.” You’re asking, “What number is that?”

Another example:

• 3y - 4 = 11
  • Variable: y
  • Given numbers: 3, 4, and 11
  • Meaning: “Three times some number, minus 4, equals 11.”

One-step vs two-step equations

• One-step equation: you only need one operation to get the variable alone.

  Examples:

  • x + 7 = 10
  • m - 3 = 9
  • 5p = 20
  • k / 4 = 2

• Two-step equation: you need two operations to get the variable alone.

  Examples:

  • 2x + 3 = 11
  • y / 5 - 4 = 2
  • 3a - 7 = 14

In each one, you can think:
“What did they do to the variable?” and “What number are we trying to reach?”

Let’s label one clearly:

• 2x + 3 = 11
  • Start: x
  • First: multiply by 2 → 2x
  • Second: add 3 → 2x + 3
  • Result must equal 11

Solving will be about undoing those actions.