# Equivalent Expressions

Consider the expressions ${3}^{2}+1$ and $5\times 2$ . Both are equal to $10$ . That is, they are equivalent expressions.

Now let us consider some expressions that include variables, say $5x+2$ .

The expression can be rewritten as $5x+2=x+x+x+x+x+1+1$ .

We can re-group the right side of the equation to $2x+3x+1+1$ or $x+4x+2$ or some other combination. All these expressions have the same value, whenever the same value is substituted for $x$ . That is, they are equivalent expressions.

Two expressions are said to be
**
equivalent
**
if they have the same value irrespective of the value of the variable(s) in them.

**
Example 1:
**

Are the two expressions $2y+5y-5+8$ and $7y+3$ equivalent? Explain your answer.

Combine the like terms of the first expression.

Here, the terms $2y$ and $5y$ are like terms. So, add their coefficients. $2y+5y=7y$ .

Also, $-5$ and $8$ can be combined to get $3$ .

Thus, $2y+5y-5+8=7y+3$ .

Therefore, the two expressions are equivalent.

**
Example 2:
**

Are the two expressions $6\left(2a+b\right)$ and $12a+6b$ equivalent? Explain your answer.

Use the Distributive Law to expand the first expression.

$\begin{array}{l}6\left(2a+b\right)=6\times 2a+6\times b\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=12a+6b\end{array}$

Therefore, the two expressions are equivalent.

**
Example 3:
**

Check whether the two expressions $2x+3y$ and $2y+3x$ equivalent.

The first expression is the sum of $2x$ 's and $3y$ 's whereas the second one is the sum of $3x$ 's and $2y$ 's.

Let us evaluate the expressions for some values of $x$ and $y$ . Take $x=0$ and $y=1$ .

$\begin{array}{l}2\left(0\right)+3\left(1\right)=0+3=3\\ 2\left(1\right)+3\left(0\right)=2+0=2\end{array}$

So, there is at least one pair of values of the variables for which the two expressions are not the same.

Therefore, the two expressions are
**
not
**
equivalent.

**
Example 4:
**

Check whether the two expressions $\frac{3\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}m\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}m}{m}$ and $m+m+m$ equivalent.

Consider the first expression for any non-zero values of the variable.

Cancel the common terms.

$\frac{3\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}m\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}\overline{)m}}{\overline{)m}}=3m$

Combine the like terms of the second expression.

$m+m+m=3m$

So, $\frac{3\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}m\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}m}{m}=m+m+m$ when $m\ne 0$ .

When $m=0$ , the expression $\frac{3\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}m\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}m}{m}$ is not defined.

That is, the expressions are equivalent except when
$m=0$
. They are
**
not
**
equivalent in general.