What does it mean when solving an equation gives a statement that is always true?
Quick answer
If simplifying an equation cancels the variable and leaves a true statement, such as 6 = 6, the original equation is an identity: every value allowed by the original equation makes it true. This differs from solving for one particular value. For example, 2x + 3 = 11 gives x = 4. Keep any restrictions from the original equation in mind; a true simplified statement does not remove them. [S1]
The question, the details, the next step.
I was solving 3(x + 2) = 3x + 6. After distributing and simplifying, I got 6 = 6. Did I make a mistake, or is there an answer?
That can be a correct result. Distribute on the left: 3(x + 2) = 3x + 6. The equation then reads 3x + 6 = 3x + 6. Subtracting 3x from both sides and then subtracting 6 leaves 6 = 6.
Because the variable terms cancel and the remaining statement is true, this equation is true for every real value of x. Its solution is all real numbers. This is called an identity. Matching variable terms can cancel, leaving a true statement such as 6 = 6, which means all real values work, subject to restrictions in the original problem. [S1]
How can I tell that this means every value works, rather than just that I happened to get a true statement while simplifying?
Each step must preserve the solutions of the original equation. In this example, distributing, subtracting the same term from both sides, and subtracting the same number from both sides are reversible steps. So the final statement describes the same solution set as the starting equation. Since 6 = 6 is true regardless of x, every allowed x works.
You can test a couple of values as a check. If x = 0, both sides are 6. If x = 5, both sides are 21. These checks illustrate the pattern, but checking individual values alone does not prove that all values work. The simplification to a true statement is what establishes the all-values result. [S1]
What would the work look like if an equation had just one solution, so I can compare the two outcomes?
For example, solve 2x + 3 = 11:
- Subtract 3 from both sides: 2x = 8.
- Divide both sides by 2: x = 4.
Here, the variable does not disappear into a statement that is true for every value. The steps isolate one value, so the solution is x = 4. Check it by substituting 4: the left side is 2(4) + 3 = 11, matching the right side. That substitution confirms that 4 works; the solved equation shows the specific value. [S1]
Could the original equation have a restriction that matters even if I end up with a true statement? What should I write in my answer?
Yes. Keep any restrictions from the original equation in view. Some expressions are only defined for certain values, so those values remain excluded even if later algebra appears to give a true statement. The all-real-numbers conclusion applies subject to restrictions in the original problem. [S1]
For 3(x + 2) = 3x + 6, there are no such restrictions, so you can write: “All real numbers; simplifying gives the true statement 6 = 6, so every real value of x satisfies the equation.” If restrictions exist, state the allowed values rather than claiming every real number. Check that each algebraic step is valid for the original expression.
A practical checklist
Simplify both sides carefully, distributing and combining like terms.
If variable terms cancel, inspect the remaining numerical statement: is it true?
If it is true, conclude that every allowed value works, not just the values you tested.
Check the original equation for restrictions on which values are allowed.
To verify a proposed single value, substitute it into the original equation and compare both sides.
Write the solution clearly: all allowed values for an identity, or the specific value for a one-solution equation.
A few more useful answers
Does checking two values prove that every value works?
No. Substitution checks only the values you try. The algebraic simplification to a true statement, using valid steps, is what shows the equation holds for every allowed value. [S1]
What should I call an equation that simplifies to 6 = 6?
You can call it an identity: the equation is true for every value allowed by the original problem. [S1]
Is “all real numbers” always the right final answer?
Only when the original equation allows every real value. If it has restrictions, report all values that satisfy the equation while respecting those restrictions. [S1]
Sources
References used for this guide.
- [S1] OpenStax Elementary Algebra 2e: General Strategy for Solving Linear Equations openstax.org · Checked