Equations with no solution
Equations with no solution arise in Grade 11 enVision Algebra 1 (Chapter 1: Solving Equations and Inequalities) when simplifying a multi-step equation produces a false numerical statement, such as 0 = −10 or 3 = 7. This happens when variable terms on both sides are identical but the constants differ, creating a contradiction that no value of the variable can satisfy. Recognizing a false statement as the signal for no solution — rather than a calculation error — is a critical algebra skill.
Key Concepts
If solving an equation with variables on both sides results in a false statement, such as $0 = 10$ or $3 = 7$, then the equation has no solution.
Common Questions
What does it mean when an equation has no solution?
It means there is no value of the variable that can make the equation true — the equation is a contradiction.
How do you know when an equation has no solution?
When you simplify the equation and the variable terms cancel, leaving a false statement like 3 = 7 or 0 = −10, the equation has no solution.
Why does a false statement indicate no solution?
A false statement is a mathematical contradiction. Since no value of the variable can turn 3 = 7 into a true statement, the original equation has no solution.
What causes an equation to have no solution?
The variable terms are identical on both sides (they cancel out) but the constant terms are different, so the equation becomes something like 5 = 8, which is always false.
How is nosolution different from a solution of 0?
A solution of x = 0 means zero satisfies the equation. No solution means the equation is impossible for any value of x.
Can you give an example of an equation with no solution?
Solve 2x + 3 = 2x − 7. Subtracting 2x from both sides gives 3 = −7, a false statement, so this equation has no solution.