Equivalent Expressions

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.

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

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 $\frac{3\times m\times m}{m}$ and $m+m+m$ equivalent.

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

Cancel the common terms.

$\frac{3\times m\times \cancel{m}}{\cancel{m}}=3m$

Combine the like terms of the second expression.

$m+m+m=3m$

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

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

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