Back

One Solution, No Solution, or Infinitely Many

SAT MathAlgebraMedium

Recognize what remains after variable terms cancel.

What you need to know

  • A true identity such as 0 = 0 means infinitely many solutions.
  • A contradiction such as 0 = 7 means no solution.
  • A statement x = number means one solution.
  • Parameter questions often ask what coefficient creates an identity or contradiction.

The AYLOTI method

  1. Expand both sides fully.
  2. Move all variable terms together.
  3. Observe whether the variable cancels.
  4. Interpret the remaining statement.

AYLOTI • Sir Hunain Zia • SAT Score 1594/1600 • 154 Total A Grades (Personal), 11 World Records, 7 Distinctions

Worked example

How many solutions does 3(x + 2) = 3x + 6 have?

  1. Expand left: 3x + 6.
  2. Both sides are identical.
  3. Every real x satisfies the equation.

Answer: Infinitely many solutions.

Common SAT traps

  • Stopping when x cancels and automatically saying no solution.
  • Assuming every linear equation has one solution.
  • Ignoring parameter values that make coefficients equal.
Quick check

How many solutions does 2x + 5 = 2x − 1 have?

No solution, because it reduces to 5 = −1.

Before you move on

  • Explain the core rule without looking back.
  • Redo the worked example using your own steps.
  • Write the trap you are most likely to fall for.
author avatar