Symmetric Property of Equality — Definition & Examples
Symmetric Property of Equality
The following property: If if a = b then b = a. This is one of the equivalence properties of equality.
See also
Reflexive property of equality, transitive property of equality, transitive property of inequalities
Key Formula
If a=b, then b=a.
Where:
- a = Any number, variable, or expression
- b = Any number, variable, or expression equal to a
Worked Example
Problem: You are given that x + 5 = 12. Use the Symmetric Property of Equality to rewrite this equation.
Step 1: Start with the given equation.
x+5=12
Step 2: Apply the Symmetric Property of Equality: if the left side equals the right side, then the right side equals the left side.
If x+5=12, then 12=x+5.
Step 3: Write the rewritten equation.
12=x+5
Answer: By the Symmetric Property of Equality, x + 5 = 12 can be rewritten as 12 = x + 5. Both forms are equally valid.
Another Example
This example shows how the property is cited as a formal justification inside a two-column or paragraph proof, rather than just rewriting a simple equation.
Problem: In a proof, you have established that 3y − 7 = 2y + 1. A later step requires the equation written with 2y + 1 on the left side. Justify the reversal.
Step 1: Write the established equation.
3y−7=2y+1
Step 2: Cite the Symmetric Property of Equality to swap the two sides.
If 3y−7=2y+1, then 2y+1=3y−7.
Step 3: State the result and the justification clearly, as you would in a formal proof.
2y+1=3y−7(Symmetric Property of Equality)
Answer: The equation is rewritten as 2y + 1 = 3y − 7, justified by the Symmetric Property of Equality.
Frequently Asked Questions
What is the difference between the Symmetric Property and the Reflexive Property of Equality?
The Reflexive Property says any quantity equals itself (a = a). The Symmetric Property says if a = b, you can reverse it to b = a. Reflexive involves one quantity; symmetric involves two quantities and lets you swap their positions.
When do you use the Symmetric Property of Equality?
You use it whenever you need to reverse the two sides of an equation. This comes up often in proofs when a previous result has the desired expression on the wrong side. It also appears in the substitution process when you need the variable isolated on the left.
Does the Symmetric Property work for inequalities?
No. If a < b, you cannot simply write b < a; that reversal is false. When you swap sides of an inequality, you must also reverse the inequality sign, giving b > a. The symmetric property applies specifically to equality, not to inequality relations.
Symmetric Property of Equality vs. Transitive Property of Equality
| Symmetric Property of Equality | Transitive Property of Equality | |
|---|---|---|
| Definition | If a = b, then b = a | If a = b and b = c, then a = c |
| Number of equalities involved | One equality, two sides swapped | Two equalities chained into a third |
| Purpose | Reverse the direction of an equation | Link two equations that share a common term |
| Example | If x = 7, then 7 = x | If x = y and y = 5, then x = 5 |
Why It Matters
The Symmetric Property of Equality appears constantly in geometry proofs, algebra proofs, and logical reasoning courses. Students are often asked to justify each step of a two-column proof, and swapping the sides of an equation requires citing this property explicitly. Understanding it also reinforces the broader idea that equality is a two-way relationship, which distinguishes it from inequalities and other non-symmetric relations.
Common Mistakes
Mistake: Applying the Symmetric Property to inequalities and writing 'if a < b then b < a.'
Correction: The Symmetric Property applies only to equality. For inequalities, swapping sides requires reversing the inequality sign: if a < b, then b > a.
Mistake: Confusing the Symmetric Property with the Commutative Property of addition or multiplication.
Correction: The Commutative Property rearranges terms within one side of an equation (a + b = b + a). The Symmetric Property swaps the entire left side with the entire right side of an equation. They are different properties.
Related Terms
- Equivalence Properties of Equality — The parent set containing the symmetric property
- Reflexive Property of Equality — States a = a; another equivalence property
- Transitive Property of Equality — Chains two equalities; another equivalence property
- Transitive Property of Inequalities — Analogous chaining rule for inequalities
- Substitution Property — Replaces one expression with an equal one
- Commutative Property — Often confused with symmetric; reorders operands