Variables in Algebraic Expressions

šŸ†Practice variables and 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 X or Y Y . This letter refers to an unknown numerical value that we must work out. For example: if X+5=8 X+5=8 , then we can conclude that the numerical value of X X is 3 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.

Labeled algebraic equation illustrating parts of an expression: terms, coefficients, variables, constants, and the full equation, using color-coded annotations for clarity.

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.

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Test yourself on variables and algebraic expressions!

einstein

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

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Examples of Algebraic Expressions

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

4+74+7

93 \frac{9}{3}

3āˆ’23-2

2Ɨ8 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 4+7 =11

93=3 \frac{9}{3}=3

3āˆ’2=1 3-2=1

2Ɨ8=16 2\times8=16


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

X+5 X + 5

Xāˆ’Y X-Y

AƗ12 A\times\frac{1}{2}

X2+6 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:

Numbers of squares n

The first figure is formed from 11 square.

The second figure is formed from 44 squares, which can be expressed as 22 by 22.

The third figure is formed from 99 squares, which can be represented as 33 by 33.

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

Answer:

n2n²


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Exercise 2

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

Exercise 2 Assignment

Solution:

In figure 1 (n=1) (n=1) there are 6āˆ’1=56-1= 5 circles.

In figure 2 (n=2) (n=2) there are 6āˆ’2=46-2=4 circles.

In figure 3 (n=3) (n=3) there are 6āˆ’3=36-3=3 circles.

In figure 4 (n=4) (n=4) there are 6āˆ’4=26-4=2 circles.

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

Answer:

6āˆ’n 6-n


Exercise 3

Simplify the following expression:

35m+9nāˆ’48m+52n 35m+9n-48m+52n

Solution:

35m+9nāˆ’48m+52n= 35m+9n-48m+52n=

First, we group like terms together.

35māˆ’48m+9n+52n= 35m-48m+9n+52n=

Then, simplify m m .

āˆ’13m+9n+52n= -13m+9n+52n=

Finally, simplify n n .

āˆ’13m+61n= -13m+61n=

61nāˆ’13m 61n-13m

Answer:

61nāˆ’13m 61n-13m


Do you know what the answer is?

Exercise 4

Simplify the following expression:

47x+57y+34x+89y\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.

47x+34x+57y+89y= \frac{4}{7}x+\frac{3}{4}x+\frac{5}{7}y+\frac{8}{9}y=

4Ɨ4+3Ɨ77Ɨ4x+5Ɨ9+7Ɨ87Ɨ9y= \frac{4\times4+3\times7}{7\times4}x+\frac{5\times9+7\times8}{7\times9}y=

16+2128x+45+5668y= \frac{16+21}{28}x+\frac{45+56}{68}y=

3728x+10168y= \frac{37}{28}x+\frac{101}{68}y=

1928x+13868y 1\frac{9}{28}x+1\frac{38}{68}y

Answer:

1928x+13868y 1\frac{9}{28}x+1\frac{38}{68}y


Exercise 5

Simplify the expression:

3baā‹…138a+58b+418m+910a+23m 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.

3baā‹…138a+58b+418m+910a+23m 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

=2baā‹…(8+3)8a+88b+910a+418m+23m =\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

=3ā‹…118ā‹…a+58b+910a+4+2ā‹…618m =\frac{3\cdot11}{8\cdot a}+\frac{5}{8}b+\frac{9}{10}a+\frac{4+2\cdot6}{18}m

=338b+58b+910a+1618m =\frac{33}{8}b+\frac{5}{8}b+\frac{9}{10}a+\frac{16}{18}m

=33+58b+910a+89m=388b+910a+89m =\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

=434b+910a+89m =4\frac{3}{4}b+\frac{9}{10}a+\frac{8}{9}m

Answer:

=434b+910a+89m =4\frac{3}{4}b+\frac{9}{10}a+\frac{8}{9}m


Check your understanding

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 XX or YY 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?

Examples with solutions for Variables and Algebraic Expressions

Exercise #1

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

Video Solution

Step-by-Step Solution

Let's simplify the expression 3x+4x+7+2 3x + 4x + 7 + 2 step-by-step:

  • Step 1: Combine Like Terms Involving x x
    The terms 3x 3x and 4x 4x are like terms because both involve the variable x x . To combine them, add their coefficients:
    3x+4x=(3+4)x=7x 3x + 4x = (3 + 4)x = 7x

  • Step 2: Combine Constant Terms
    The expression includes constant terms 7 7 and 2 2 . These can be added together to simplify:
    7+2=9 7 + 2 = 9

  • Step 3: Write the Simplified Expression
    Now, combine the results from Step 1 and Step 2 to form the final simplified expression:
    7x+9 7x + 9

Therefore, the simplified expression is 7x+9 7x + 9 .

Reviewing the choices provided, the correct choice is:

  • Choice 2: 7x+9 7x + 9

This matches our simplified expression, confirming our solution is correct.

Answer

7x+9 7x+9

Exercise #2

3z+19zāˆ’4z=? 3z+19z-4z=\text{?}

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Combine like terms by identifying and adding their coefficients.
  • Step 2: Simplify the expression.
  • Step 3: Verify the resulting expression with the provided choices.

Let's work through each step:

Step 1: Identify the coefficients in the expression 3z+19zāˆ’4z 3z + 19z - 4z . The coefficients are 3 3 , 19 19 , and āˆ’4 -4 .

Step 2: Add and subtract these coefficients: 3+19āˆ’4 3 + 19 - 4 .

Step 3: Calculate: 3+19=22 3 + 19 = 22 and then 22āˆ’4=18 22 - 4 = 18 .

Therefore, the simplified expression is 18z 18z .

The solution to the problem is 18z 18z .

Answer

18z 18z

Exercise #3

Are the expressions the same or not?

20x 20x

2Ɨ10x 2\times10x

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Simplify the expression 2Ɨ10x 2 \times 10x .
  • Step 2: Compare the simplified expression with 20x 20x .

Now, let's work through each step:
Step 1: The expression 2Ɨ10x 2 \times 10x can be rewritten using associativity as 2Ɨ(10Ɨx) 2 \times (10 \times x) .
Step 2: Apply the associative property of multiplication: (2Ɨ10)Ɨx=20Ɨx=20x (2 \times 10) \times x = 20 \times x = 20x .

Comparing this with the given expression, we see that both expressions are indeed the same, as they simplify to 20x 20x .

Therefore, the solution to the problem is Yes.

Answer

Yes

Exercise #4

Are the expressions the same or not?

3+3+3+3 3+3+3+3

3Ɨ4 3\times4

Video Solution

Step-by-Step Solution

To solve this problem, we'll analyze the expressions 3+3+3+33+3+3+3 and 3Ɨ43 \times 4 to determine if they are equivalent.

First, evaluate the expression 3+3+3+33+3+3+3:

  • Add the numbers: 3+3=63 + 3 = 6
  • Add again: 6+3=96 + 3 = 9
  • Add the last 33: 9+3=129 + 3 = 12

The result of 3+3+3+33+3+3+3 is 1212.

Next, evaluate the expression 3Ɨ43 \times 4:

  • Perform the multiplication: 3Ɨ4=123 \times 4 = 12

The result of 3Ɨ43 \times 4 is also 1212.

Since both expressions result in the same number, we conclude that

The expressions are the same.

Therefore, the correct answer is Yes.

Answer

Yes

Exercise #5

Are the expressions the same or not?

18x 18x

2+9x 2+9x

Video Solution

Step-by-Step Solution

To determine if the expressions 18x 18x and 2+9x 2 + 9x are equivalent, we'll analyze their structures.

  • 18x 18x is a linear expression with a single term involving the variable x x , and its coefficient is 18.
  • 2+9x 2 + 9x consists of two terms: a constant term 2 2 and a linear term 9x 9x with coefficient 9.

For two expressions to be equivalent, each corresponding term must be equal. Here, the expression 18x 18x has no constant term, whereas 2+9x 2 + 9x has a constant term of 2. Furthermore, the linear term coefficients differ: 18≠9 18 \neq 9 .

Therefore, the expressions 18x 18x and 2+9x 2 + 9x are not the same. They structurally differ and cannot be made equivalent just through similar values of x x .

Therefore, the solution to this problem is: No.

Answer

No

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