Identity and No-Solution Equations

Key idea

Some equations are always true or never true.

In depth

If solving gives a true statement like 5 = 5, every value works (identity). If it gives a false one like 3 = 7, no value works. Otherwise there is exactly one solution.

Worked example

2(x+1) = 2x + 2 → 0 = 0, infinitely many solutions. 2x + 1 = 2x + 5 → 1 = 5, no solution.

Real-world uses

• Checking whether two formulas are the same
• Spotting impossible results in a model

Common mistake

Saying x = 0 when the variable cancels out.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related intermediate topics in Algebra

• Multi-Step Equations – Distribute, combine like terms, move variables to one side.
• Variables on Both Sides – Collect variable terms on one side first.
• Literal Equations – Rearrange formulas to solve for a chosen variable.
• Linear Inequalities – Solve like equations but reverse the sign when multiplying by a negative.
• Compound Inequalities – Combine two inequalities using “and” or “or”.
• Absolute Value Equations – Split into two cases: inside equals ± the value.