Transitive Property of Equality - Definition & Examples

Transitive Property of Equality

The Transitive Property of Equality is one of the math Properties of Equality, including the Reflexive Property, the Symmetric Property, the Addition and Subtraction Properties of Equality, and the Multiplication and Division Properties of Equality.

The Transitive Property of Equality states: $$\text{If $a=b$ and $b=c,$ then $a=c$}$$

Why Does This Hold?

Let's say that we have the following equalities:

$$\begin{align} a&=b\\ b&=c \end{align}$$

By the Symmetric Property of Equality, we know that writing $a=b$ is the same as writing $b=a.$ Now we have two new equalities: $$\begin{align} b&=a\\ b&=c \end{align}$$

But wait! Now $b$ equals both $a$ and $c!$ Since $b$ cannot equal two completely different quantities, $a$ and $c$ must be equal. So, $$a=c$$

Let's look at a few examples of the Transitive Property of Equality.

Example One

First, consider the following equalities. $$\begin{align} x&=3^2\\ 3^2&=9 \end{align}$$ By the Transitive Property of Equality, we know that if $x=3^2$ and $3^2=9,$ then $x=9$.

Let's assign variables to the heights of Kami, Andre, and Gwen to simplify the problem. $$\begin{align} k &= \text{Kami's height}\\ a &= \text{Andre's height}\\ g &= \text{Gwen's height} \end{align}$$

The problem says that Kami is the same height as Andre, and Andre is the same height as Gwen. Thus, $$\begin{align} k&=a\\ a&=g \end{align}$$

We know by the Transitive Property of Equality that if $k=a$ and $a=g,$ then $k=g.$ So, Kami's height is the same as Gwen's height. Since Kami is $6$ feet tall, Gwen must also be $6$ feet tall.

Transitive Property of Equality

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

Here is a graphic stating the Transitive Property of Equality:

For example: $$\text{If } x=y\text{ and } y=z,\\ \text{then }x=z$$

$$\text{If } s=2\text{ and } 2=t,\\ \text{then }s=t$$ $$\text{If } p+q=r^2\text{ and } r^2=15,\\ \text{then }p+q=15$$

Here's a way to visualize why the Transitive Property of Equality works:

The transitive property of equality states that: $$\text{if }a = b\text{ and }b=c\text{, then }a=c$$

Let's see some examples to get a better sense of this.

1. if $x = y$ and $y=3$, then $x=3$
2. if $x = y$ and $y=z$, then $x=z$
3. if $x = y+5$ and $y+5=3z$, then $x=3z$

These are also equivalent equations!

Now, try the problem below to test your understanding.

Now, a slightly harder question...

You'll use the transitive property when you've evaluating functions and graphing!

(Video) What is the Transitive Property of Equality?

by Tarver Academy

This video by Tarver Academy explains the Transitive Property of Equality.

$\large{\text{Summary}}$

The Transitive Property of Equality states: $$\text{If $a=b$ and $b=c,$ then $a=c$}$$

Here are some examples:

• If $x=7$ and $7=y,$ then $x=y$.
• If $t=17$ and $17=x+3,$ then $t=x+3$.
• If $x+y=z+w$ and $z+w=a+b,$ then $x+y=a+b$.