Special Cases: Identities and No-Solution Equations
Grade 9 students in California Reveal Math Algebra 1 learn to recognize the three possible outcomes when solving a linear equation: one solution, infinitely many solutions (identity), or no solution (contradiction). An identity occurs when all variables cancel and a true statement like 3=3 remains — meaning the equation is satisfied by every real number. A contradiction occurs when variables cancel and a false statement like 0=5 remains — meaning no value satisfies the equation. For example, 2(x+3)=2x+6 is an identity (true for all x), while 3x+4=3x-2 is a contradiction (no solution).
Key Concepts
When solving a linear equation, one of three outcomes is possible:.
One Solution: The variable isolates to a single value, e.g., $x = a$.
Common Questions
What are the three outcomes possible when solving a linear equation?
One solution (the variable isolates to a single value), infinitely many solutions / identity (all variables cancel leaving a true statement), or no solution / contradiction (all variables cancel leaving a false statement).
What is an identity equation?
An identity is an equation where all variable terms cancel and a true statement remains, like 3=3. The equation is true for every real number, so it has infinitely many solutions.
What is a contradiction equation?
A contradiction is an equation where all variable terms cancel and a false statement remains, like 0=5. No value of the variable satisfies the equation.
Is 2(x+3)=2x+6 an identity or contradiction?
Identity. Distributing gives 2x+6=2x+6. Subtracting 2x from both sides: 6=6, which is true. The equation is satisfied by all real numbers.
Is 3x+4=3x-2 an identity or contradiction?
Contradiction. Subtracting 3x from both sides: 4=-2, which is false. No value of x makes this true — no solution.
Which unit covers identity and no-solution equations in Algebra 1?
This skill is from Unit 1: Using Expressions and Equations in California Reveal Math Algebra 1, Grade 9.