Variables in Algebraic Expressions

When a problem is presented to us in writing, we can convert it into mathematical language (also called algebraic language) by transforming it into an algebraic expression. But what are algebraic expressions?

Variable: This is a letter that represents a numerical value, for example $X$ or $Y$ . This letter refers to an unknown numerical value that we must work out. For example: if $X+5=8$ , then we can conclude that the numerical value of $X$ is $3$ .

An algebraic expression is a combination of numbers and letters (representing unknown numbers) that includes operations such as addition, subtraction, multiplication, division, etc.

Each element of an algebraic expression is called an algebraic term, be it a variable, a constant, or a combination of a coefficient and one or more variables. If the expression contains only one term, it is known as a monomial, while those that contain two or more terms are polynomials.

There is no limitation to the amount of constant numbers, unknown variables, or operations that can appear in an algebraic expression. In addition, there does not always have to be a variable in the algebraic expression, although it will always have a certain numerical value.

Detailed explanation

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Test yourself on algebraic expressions!

Are the expressions the same or not?

$$ 20x $$

$2\times10x$

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Examples of Algebraic Expressions

Let's take a look at some examples of algebraic expressions without variables:

$4+7$

$\frac{9}{3}$

$3-2$

$2\times8$

Here we can see that all of the expressions are composed of numbers and, since there are no unknown variables, we can calculate the result by simply performing the operations.

$4+7 =11$

$\frac{9}{3}=3$

$3-2=1$

$2\times8=16$

Now, let's look at some examples with variables:

$X + 5$

$X-Y$

$A\times\frac{1}{2}$

$X^2+6$

In this case, the examples include numbers, unknown variables (represented by letters), and mathematical operations (addition, subtraction, multiplication, division, etc.).

Exercises: Variables in Algebraic Expressions

Exercise 1

Find the algebraic expression that corresponds to the number of squares in the nth figure.

algebraic expressions corresponding to the number of squares

Solution:

The first figure is formed from $1$ square.

The second figure is formed from $4$ squares, which can be expressed as $2$ by $2$ .

The third figure is formed from $9$ squares, which can be represented as $3$ by $3$ .

Following this pattern, we can work out that the nth figure will be formed from $n \times n = n²$ squares.

Answer:

$n²$

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Test your knowledge

Question 1

Are the expressions the same or not?

$$ 3+3+3+3 $$

$3\times4$

Question 2

Are the expressions the same or not?

$$ 18x $$

$$ 2+9x $$

Question 3

$$ x+x= $$

Exercise 2

Find the algebraic expression that describes the number of circles in the figure $n$ .

Solution:

In figure 1 $(n=1)$ there are $6-1= 5$ circles.

In figure 2 $(n=2)$ there are $6-2=4$ circles.

In figure 3 $(n=3)$ there are $6-3=3$ circles.

In figure 4 $(n=4)$ there are $6-4=2$ circles.

Following this pattern, we can work out that there will be $6-n$ circles in the nth figure.

Answer:

$6-n$

Exercise 3

Simplify the following expression:

$35m+9n-48m+52n$

Solution:

$35m+9n-48m+52n=$

First, we group like terms together.

$35m-48m+9n+52n=$

Then, simplify $m$ .

$-13m+9n+52n=$

Finally, simplify $n$ .

$-13m+61n=$

$61n-13m$

Answer:

$61n-13m$

Do you know what the answer is?

Question 1

$$ 5+8-9+5x-4x= $$

Question 2

$$ 5+0+8x-5= $$

Question 3

$$ 11+5x-2x+8= $$

Exercise 4

Simplify the following expression:

$\frac{4}{7}x+\frac{5}{7}y+\frac{3}{4}x+\frac{8}{9}y$

Solution:

The like terms are grouped together and the fraction operations are performed.

$\frac{4}{7}x+\frac{3}{4}x+\frac{5}{7}y+\frac{8}{9}y=$

$\frac{4\times4+3\times7}{7\times4}x+\frac{5\times9+7\times8}{7\times9}y=$

$\frac{16+21}{28}x+\frac{45+56}{68}y=$

$\frac{37}{28}x+\frac{101}{68}y=$

$1\frac{9}{28}x+1\frac{38}{68}y$

Answer:

$1\frac{9}{28}x+1\frac{38}{68}y$

Exercise 5

Simplify the expression:

$3\frac{b}{a}\cdot1\frac{3}{8}a+\frac{5}{8}b+\frac{4}{18}m+\frac{9}{10}a+\frac{2}{3}m$

Solution:

Here, the multiplication is performed and then the like terms are simplified.

$3\frac{b}{a}\cdot1\frac{3}{8}a+\frac{5}{8}b+\frac{4}{18}m+\frac{9}{10}a+\frac{2}{3}m$

$=\frac{2b}{a}\cdot\frac{\left(8+3\right)}{8}a+\frac{8}{8}b+\frac{9}{10}a+\frac{4}{18}m+\frac{2}{3}m$

$=\frac{3\cdot11}{8\cdot a}+\frac{5}{8}b+\frac{9}{10}a+\frac{4+2\cdot6}{18}m$

$=\frac{33}{8}b+\frac{5}{8}b+\frac{9}{10}a+\frac{16}{18}m$

$=\frac{33+5}{8}b+\frac{9}{10}a+\frac{8}{9}m=\frac{38}{8}b+\frac{9}{10}a+\frac{8}{9}m$

$=4\frac{3}{4}b+\frac{9}{10}a+\frac{8}{9}m$

Answer:

$=4\frac{3}{4}b+\frac{9}{10}a+\frac{8}{9}m$

Check your understanding

Question 1

$3x+4x+7+2=\text{?}$

Question 2

$3z+19z-4z=\text{?}$

Question 3

Are the expressions the same or not?

$0.5x\times1$

$$ 0.5x+0 $$

Review Questions

What is a 'variable' in mathematics?

A variable is an unknown number.

How is a variable represented?

Variables are represented by a symbol, usually a letter of the alphabet such as $X$ or $Y$ although Greek letters are also often used.

Are there any other names for variables?

Yes, sometimes they are also referred to as 'unknowns' or 'literals'.

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

On the Tutorela website you will find a variety of other useful mathematics articles!

How many exercises should I practice?

Since each student learns at a different pace, the answer to this question depends on you.
The important thing is that you are aware of your level and therefore whether or not you need to practice the formulas more.
That said, it is recommended that you do 10 basic and intermediate level exercises in order to learn a single basic formula.

Do you think you will be able to solve it?

Question 1

Are the expressions the same or not?

$$ 15x-30 $$

$$ 45-15-5x+15x $$

Question 2

$7a+8b+4a+9b=\text{?}$

Question 3

$18x-7+4x-9-8x=\text{?}$

Examples with solutions for Algebraic Expressions

Exercise #1

$18x-7+4x-9-8x=\text{?}$

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

$14x-16$

Exercise #2

$8y+45-34y-45z=\text{?}$

Video Solution

Step-by-Step Solution

To solve this question, we need to remember that we can perform addition and subtraction operations when we have the same variable,
but we are limited when we have several different variables.

We can see in this exercise that we have three variables:
$45$ which has no variable
$8y$ and $34y$ which both have the variable $y$
and $45z$ with the variable $z$

Therefore, we can only operate with the y variable, since it's the only one that exists in more than one term.

Let's rearrange the exercise:

$45-34y+8y-45z$

Let's combine the relevant terms with $y$

$45-26y-45z$

We can see that this is similar to one of the other answers, with a small rearrangement of the terms:

$-26y+45-45z$

And since we have no possibility to perform additional operations - this is the solution!

Answer

^{$-26y+45-45z$}

Exercise #3

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

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

$37.2a+2.3$

Exercise #4

$\frac{8x^2}{4x}+3x=$

Video Solution

Step-by-Step Solution

Let's break down the fraction's numerator into an expression:

$8x^2=4\times2\times x\times x$

And now the expression will be:

$\frac{4\times2\times x\times x}{4x}+3x=$

Let's reduce and get:

$2x+3x=5x$

Answer

$5x$

Exercise #5

Simplifica la expresión:

$2x^3\cdot x^2-3x\cdot x^4+6x\cdot x^2-7x^3\cdot 5=$

Video Solution

Step-by-Step Solution

We'll use the law of exponents for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

We'll apply this law to the expression in the problem:

$2x^3\cdot x^2-3x\cdot x^4+6x\cdot x^2-7x^3\cdot 5=2x^{3+2}-3x^{1+4}+6x^{1+2}-35x^3$

When we apply the above law to the first three terms from the left, while remembering that any number can always be considered as that number raised to the power of 1:

$a=a^1$

And in the last term we performed the numerical multiplication,

We'll continue and simplify the expression we got in the last step:

$2x^{3+2}-3x^{1+4}+6x^{1+2}-35x^3=2x^5-3x^5+6x^3-35x^3=-x^5-29x^3$

Where in the first stage we simplified the expressions in the exponents of the terms in the expression and in the second stage we combined like terms,

Therefore the correct answer is answer A.

Answer

$-x^5-29x^3$

Start practice

Variables in Algebraic Expressions

Test yourself on algebraic expressions!

Examples of Algebraic Expressions

Exercises: Variables in Algebraic Expressions

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Review Questions

Examples with solutions for Algebraic Expressions

Exercise #1

Video Solution

Step-by-Step Solution

Answer

Exercise #2

Video Solution

Step-by-Step Solution

Answer

Exercise #3

Video Solution

Step-by-Step Solution

Answer

Exercise #4

Video Solution

Step-by-Step Solution

Answer

Exercise #5

Video Solution

Step-by-Step Solution

Answer

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Variables and Algebraic Expressions