Equivalent Expressions

Here are some examples:

• $1+1=2$
• $2-0=2$
• $7X=2X+5X$
• $4X\times\left(2+3\right)=8X+12X$
• $8X+12X=20X$

Write an equivalent expression for the following:

$0$

Solution

We look for an expression that represents $0$, for example:

$0=5-5$

$3+3+3$

Solution

To do this exercise, we must first work out that the expression represents $9$ before looking for an equivalent form.

$3+3+3=10-1$

$7X$

Solution

We look for a way to represent $7X$, e.g.:

$7x=4x+2x+x$

$13X−3$

Solution

We look for an equivalent way of representing $13X$ and $-3$, for instance:

$13x-3=15x-2x-2-1$

$1.5X+8+6.5X$

Solution

As the expression represents $8X+8$, we need to look for an alternative way to represent each term:

$1.5x+8+6.5x=10x-2x+5+3$

$18X$

$2+9X$

Solution

The expressions are not equivalent. One represents $18X$ while the other one represents $9X$.

$20X$

$2\times10X$

Solution

The expressions are equivalent as both represent $20X$.

$3+3+3+3$

$3\times4$

Solution

The expressions are equivalent because both represent the number $12$.

$15X−30$

$45-15-5X+15X$

Solution

The expressions are not equivalent. The first one represents $15X$ while the second one represents only $10X$.

$0.5X\times1$

$0.5X+0$

Solution

The expressions are equivalent.

If you are interested in this article, you may also be interested in the following articles:

• Unknowns and algebraic expressions.
• Numerical Value in Algebraic Expressions
• Multiplication of Algebraic Expressions
• Simplification of elements / Simplification of similar elements

On Tutorela website you will find a wide range of useful mathematics articles!

What is an algebraic expression?

An algebraic expression is a combination of numbers, letters (representing unknown numbers), and arithmetic operations.

What are equivalent algebraic expressions?

They are algebraic expressions that have different structures but represent the same value.

How to find equivalent expressions?

We try to modify the structure of the expression so that the value it represents is not altered.

It is important that we learn to write equivalent algebraic expressions in their simplest form since this will be very useful when solving equations.

Simplify $2x+5x+x$

Solution:

To find an equivalent expression we add the coefficients of each term together.

$\left(2+5+1\right)x=8x$

Reduce the expression $8m+8-6m+3$

Solution:

We separate the terms that have $m$ from those that do not and perform the indicated operations.

$8m+8-6m+3=8m-6m+8+3=(8-6)m+11=2m+11$

Find a simpler equivalent form for the expression $8x+2y-3z+3y-4x+3z$

Solution:

We first group the terms that have the same letter and then perform the indicated operations.

$8x+2y-3z+3y-4x+3z=8x-4x+2y+3y-3z+3z=(8-4)x+\left(2+3\right)y+\left(-3+3\right)z=4x+5y+0z=4x+5y$

Simplify the expression $6x+1-2x+3$. Then substitute the value x=3 in to both expressions and verify that you get the same numerical value.

Solution:

First we simplify the expression.

$6x+1-2x+3=6x-2x+1+3=4x+4$

Now we substitute $x=3$ in to both expressions.

$6\left(3\right)+1-2\left(3\right)+3=18+1-6+3=18+1+3-6=22-6=16$

$4\left(3\right)+4=12+4=16$

We do indeed get the same numerical value from both expressions.

Solve the equation $5x+2+3x+7-2x-5=16$

Solution:

First, find an equivalent expression:

$5x+2+3x+7-2x-5=16$

$5x+3x-2x+2+7-5=16$

$\left(5+3-2\right)x+4=16$

$6x+4=16$

Now solve for $x$:

$6x=16-4$

$6x=12$

$x=\frac{12}{6}$

$x=2$