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Symmetric Property of Equality

Math is made up of a bunch of properties that allow you to do cool things with math. One set of properties are the Algebraic Properties we saw in pre-algebra, like the Associative Property and the Commutative Property.

A second set of properties are the Properties of Equality. The Symmetric Property of Equality, along with the Reflexive and Transitive Properties of Equality, are some of the most important properties in math.

In other words, if you have an equal sign between two values, or variables, these values are equal, no matter which side of the sign they are on.

For example: $$\text{If } x=10,\\ \text{then }10=x$$

$$\text{If } y=z^2,\\ \text{then }z^2=y$$ $$\text{If } s=t+1,\\ \text{then }t+1=s$$

In each of these equations, the expression on one side of the equal sign is identical to the expression on the other side of the equal side.

One way to remember the Symmetric Property of Equality is to think about the words symmetric or symmetrical. If you fold a piece of paper in half, and both halves have the exact same shape, then the piece of paper is symmetrical. Each half is the mirror image of the other half.

Take a look at this picture. Notice that if you folded across the dotted line, each equation is the mirror image of the other equation. They're symmetrical!

This idea that both sides of the equation must be equal is also seen with the Addition and Subtraction Properties of Equality and the Multiplication and Division Properties of Equality.

This property will be incredibly helpful when you're evaluating functions and when you're estimating values for functions.

The symmetric property is straightforward. In equations, what is on which side does not matter. That means, if $a$ equals $b$, then $b$ equals $a$. $$\begin{align} \text{if }\,a&=b\\ \text{then }\,b&=a \end{align}$$

Here are some examples:

$$\begin{array}{cc} a)& \begin{gathered} 5+5 = 10 \\ 10 = 5+5 \end{gathered}\\ \\ b)& \begin{gathered} x+5 = 10 \\ 10 = x+5\end{gathered}\\ \\ c)& \begin{gathered} x+y = w+z\\ w + z = x+y \end{gathered} \end{array}$$

When you have a equal sign, feel free to swap the left and right side as you like!

(Video) What is the Symmetric Property of Equality?

by Tarver Academy

In this video, Tarver Academy explains the Symmetric Property of Equality. This property states that if $a=b,$ then $b=a.$

$\large{\text{Summary}}$

The symmetric property of equality is fairly simple. $$\begin{gather} \text{If }A=B\\ \text{then, }B=A \end{gather}$$

So if $x=3$ then $3=x$.

If we apply this to inequalities we just have to flip the sign. $$\begin{gather} \text{If }x<y\\ \text{then, }y>x \end{gather}$$