Variables in 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.).

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

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:

$n²$

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:

$6-n$

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:

$61n-13m$

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:

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

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:

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

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:

• The numerical value in algebraic expressions.
• Equivalent Expressions / Equivalent Algebraic Expressions
• Multiplication of Algebraic Expressions
• Simplification of elements / Simplification of similar elements

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.