**Here are some examples:**

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

## Practicing Equivalent Expressions

### Exercise 1

**Write an equivalent expression for the following:**

$0$

**Solution**

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

$0=5-5$

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### Exercise 2

$3+3+3$

**Solution**

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

$3+3+3=10-1$

### Exercise 3

$7X$

**Solution**

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

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

Do you know what the answer is?

### Exercise 4

$13X−3$

**Solution**

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

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

### Exercise 5

$1.5X+8+6.5X$

**Solution**

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

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

## Which of the following expressions are are equivalent?

### Exercise 6

$18X$

$2+9X$

**Solution**

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

### Exercise 7

$20X$

$2\times10X$

**Solution**

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

Do you think you will be able to solve it?

### Exercise 8

$3+3+3+3$

$3\times4$

**Solution**

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

### Exercise 9

$15X−30$

$45-15-5X+15X$

**Solution**

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

### Exercise 10

$0.5X\times1$

$0.5X+0$

**Solution**

The expressions are equivalent.

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## Questions and Answers: Equivalent Expressions

**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.

Do you know what the answer is?

## Additional Examples

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

### Exercise 1

**Simplify** **$2x+5x+x$**

**Solution:**

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

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

### Exercise 2

**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.

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

### Exercise 3

**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.

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

Do you think you will be able to solve it?

### Exercise 4

**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.

### Exercise 5

**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 $x$:

$6x=16-4$

$6x=12$

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

$x=2$

## Examples with solutions for Equivalent Expressions

### Exercise #1

### Video Solution

### Step-by-Step Solution

To solve the exercise, we will reorder the numbers using the substitution property.

$18x-8x+4x-7-9=$

To continue, let's remember an important rule:

1. It is impossible to add or subtract numbers with variables.

That is, we cannot subtract 7 from 8X, for example...

We solve according to the order of arithmetic operations, from left to right:

$18x-8x=10x$$10x+4x=14x$$-7-9=-16$Remember, these two numbers cannot be added or subtracted, so the result is:

$14x-16$

### Answer

### Exercise #2

$7.3\cdot4a+2.3+8a=\text{?}$

### Video Solution

### Step-by-Step Solution

It is important to remember that when we have numbers and variables, it is impossible to add or subtract them from each other.

We group the elements:

$7.3×4a + 2.3 + 8a =$

29.2a + 2.3 + 8a =

$37.2a + 2.3$

And in this exercise, this is the solution!

You can continue looking for the value of a.

But in this case, there is no need.

### Answer

### Exercise #3

$\frac{9m}{3m^2}\times\frac{3m}{6}=$

### Video Solution

### Step-by-Step Solution

According to the laws of multiplication, we must first simplify everything into one exercise:

$\frac{9m\times3m}{3m^2\times6}=$

We will simplify and get:

$\frac{9m^2}{m^2\times6}=$

We will simplify and get:

$\frac{9}{6}=$

We will factor the expression into a multiplication:

$\frac{3\times3}{3\times2}=$

We will simplify and get:

$\frac{3}{2}=1.5$

### Answer

### Exercise #4

### Video Solution

### Answer

### Exercise #5

### Video Solution

### Answer