Use of Variables in Common Rules: Algebra is a branch of Mathematics that deals with expressions, equations that contain variables, and constants. Variables in Algebra are the unknown value of a quantity or value in the equation. The variable does not have a fixed value. We have different types of uses of variables in different branches of Mathematics.

Variables are used in the common rules of Algebra, such as perimeter and area in Geometry and laws such as commutative, associative and distributive in arithmetic. Variables are also important in formulating and generalising the expressions, equations and identities.

What is a Variable?

Variable is a quantity that can be altered to solve a mathematical problem. A variable, in other words, is a symbol for a number whose value is unknown. A variable in algebra is represented using letters of the English alphabet. The generic letters \(x,\,y\), and \(z\) are used frequently in algebraic expressions and equations. Variables that are commonly used include \(x\) and \(y\) (real-number unknowns), \(z\) (complex-number unknowns), \(t\) (time), \(r\) (radius), and \(s\) (arc length).

Example:
\(x – 5 = 0\)
By solving the equation, you can easily find the value of the variable \(x\). If the equation is solved in this case, the value of the variable \(x\) is obtained as \(5\). It follows that \(x = 5\).
The term variable is also used in Statistics. A variable is also known as a data item in Statistics. It denotes the number of variables that can be measured. For example, variables in Statistics include gender, age, income, and capital expenditure.

Use of Variables to Find the Perimeter

The perimeter of the plane figure is the sum of the length of the sides of a figure. The variable represents the length of the sides a plane figure. By using this variable, we will find the perimeter of the figure.

Shape	Perimeter Formula	Figures
Equilateral triangle	\(3l\), where \(l\) is the length of each side of the triangle
Square	\(4l\), where \(l\) is the length of each side of the square
Regular pentagon	\(5l\), where \(l\) is the length of each side of the pentagon
Rectangle	\(2\left( {l + b} \right)\), where length and breadth of rectangle is represented by variables \(l\) and \(b\) respectively.

Thus, variables are used in finding the perimeter of closed figures.

Use of Variables in Finding the Area

There are different formulas associated with finding the area of the different polygons, where the lengths of sides of the polygons are represented by using the variables.

Shape	Formula	Notes
Right triangle	\(\frac{1}{2} \times b \times h\)	Base and height of the triangle are represented by the variables \(b\) and \(h\)
Equilateral triangle	\(\frac{{\sqrt 3 }}{4}{l^2}\)	Length of each side of the equilateral triangle is represented by the variable \(l\)
Square	\({l^2}\)	Length of each side of the square is represented by the variable \(l\)
Rectangle	\(2\left( {l + b} \right)\)	The length and breadth of the rectangle are denoted by the variables \(l\) and \(b\).

Use of Variables in Finding the Number Patterns

Various patterns such as successor and predecessor of number, even and odd numbers are represented as formulas in general by using the variables.

• The successor of any natural number \(n,\)
  • Successor is given by \(n + 1.\)
  • Predecessor is given by \(n – 1.\)
• Square of any natural number \(n\), is denoted by \({n^2}\).
• Cube of any natural number \(n\), is denoted by \({n^3}\).
• For any natural number \(n\),
  • Even number is given by \(2n\)
  • Odd number is given by \(2n + 1\).

Note: Not only natural numbers but any number, such as integers and rational numbers, are also shown by using the variables \(Z\) and \(R\), respectively.

Use of Variables in Geometry

We know that variables are used in geometry to find area and perimeter. Some of the other rules where we can use the variables are given below.

• Express the value of the diameter of a circle in terms of radius, which is given by \(d = 2r\), where \(d\)-diameter and \(r\)-radius of the circle.
• The number of diagonals of the plane figure that can be drawn from the one vertex is given by \(\left( {n – 3} \right)\), where the number of sides of the polygon is represented by \(n.\).
• The angle sum property of a polygon of side \(n\) is given by \({180^ \circ } \times \left( {n – 2} \right)\), where the number of sides of the polygon is represented by \(n\).

Use of Variables in Arithmetic

In arithmetic, variables play an important role in representing unknown values or describing a property.

Commutative Property under Addition

We know that \(2 + 3 = 5\), and it can be written as \(3 + 2 = 5\)
Here, \(2 + 3 = 3 + 2\)
This property of numbers is called the commutative property under addition. When we add two numbers, the resultant value is the same as when the order of the numbers is changed. Thus, commutating means changing the order. This property holds for any number.
Let two numbers be represented by variables \(a\) and \(b\), then
\(a + b = b + a\)

Commutative Property under Multiplication

We know that \(3 \times 5 = 15\) and also \(5 \times 3 = 15\).
Here, \(3 \times 5 = 5 \times 3\)
This property of numbers is called the commutative property of numbers under multiplication. When two numbers are multiplied together, the product obtained and the product obtained by changing the order of the numbers is the same. This property holds for any number.
Let two numbers are represented by variables \(a\) and \(b\), then
\(a \times b = b \times a\)

Distributive Property

To find the value of \(2 \times 23\), we can write as
\(2 \times \left( {20 + 3} \right)\)
\(2 \times 20 + 2 \times 3\)
And, by direct multiplication, \(2 \times 23 = 46\)
Thus, \(2 \times 23 = 2 \times \left( {20 + 3} \right)\)
This property of numbers is called the distributive property under addition. This property holds for any three numbers. If any three numbers are represented as \(a,b\) and \(c\), then
\(a \times \left( {b + c} \right) = \left( {a \times b} \right) + \left( {a \times c} \right)\)

Associative Property under Addition

Let us considers any three numbers \(2,3,\) and \(5.\) Then,
\(\left( {2 + 3} \right) + 5 = 5 + 5 = 10\)
\(2 + \left( {3 + 5} \right) = 2 + 8 = 10\)
This property of numbers is called the associative property under addition. If three numbers are represented as \(a,b\) and\(c\), then
\(a + \left( {b + c} \right) = \left( {a + b} \right) + c\)

Associative Property under Multiplication

Let us considers any three numbers \(3,5,\) and \(7\). Then,
\(\left( {3 \times 5} \right) \times 7 = 15 \times 7 = 105\)
\(3 \times \left( {5 \times 7} \right) = 3 \times 35 = 105\)
This property of numbers is called the associative property under multiplication. This property holds for any three numbers. If three numbers are represented as \(a,b\) and \(c\), then
\(a \times \left( {b \times c} \right) = \left( {a \times b} \right) \times c\)

Use of Variables in Algebra

Variable is the unknown quantity of the number in a given equation. Algebra is the branch of mathematics that deals with expressions and equations. Some of the uses of the variables in algebra are listed below.

• Represent the expressions and equations
  Example: \(x + 2,\,\,2x + 5 = 0\)
• Denote the standard polynomials and equations
  Example: \(a{x^2} + bx + c = 0\) is the quadratic equation
• Representing the arithmetic and geometric sequences.
  Example: \(a,a + d,a + 2d,a + 3d, \ldots \) is called A.P.
• Formulating the identities
  Example: \({a^2} – {b^2} = \left( {a + b} \right)\left( {a – b} \right)\)

Solved Examples – Use of Variables in Common Rules

Below are a few solved examples that can help in getting a better idea.

Q.1. As illustrated in the figure, a cube is a three-dimensional figure. It has six faces that are all equivalent squares. The length of a cube’s edge is given by \(l\). Compute the formula for the total length of the cube’s edges.
Ans: Given: Length of each edge of a cube is \(l\) units.

In a cube, all edges of the cube are equal in length. So, the length of all edges of the cube is \(l\) units. There are a total \(12\) edges in the cube.
Total length of the edges of a cube is the sum of the lengths of all the edges.
Thus, total length of edges is \(12\,l\) units.
There are a total \(12\) edges in the cube.
Total length of the edges of a cube is the sum of the lengths of all the edges.
Thus, total length of edges is \(12\,l\) units.

Q.2. The side of the regular hexagon is \(l\) units. Then, express the perimeter of a regular hexagon by using the given variable.
Ans: Given: Length of the side of the regular hexagon is \(l\) units.

We know that a regular hexagon is a hexagon with all six sides equal.
So, the length of all sides is \(l\) units.
Perimeter of the regular hexagon is the length of the boundary, which is equal to the length of all sides.
Therefore, the perimeter of a regular hexagon is \(6\,l\) units.

Q.3. As shown in the below figure, \(C\) is the centre of a circle, and \(AB\) is the diameter of the circle. Let \(r\) be the radius of the circle. Express the diameter in terms of radii of the circle by using the given variable.

Ans:

Given: \(C\) is the centre of a circle and \(AB\) is the diameter of the circle, and \(r\) is the radius of the circle.
Diameter of a circle is twice the length of the radius of the circle.
Diameter \( = 2 \times {\text{radius}}\)
\(AB = 2r\)
The above equation gives the expression of diameter in terms of radius.

Q.4. List any five uses of variables in algebraic identities.
Ans: Some of the algebraic identities are
1. \({\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\)
2. \({\left( {a – b} \right)^2} = {a^2} – 2ab + {b^2}\)
3. \({a^2} – {b^2} = \left( {a + b} \right) \times \left( {a – b} \right)\)
4. \({a^3} – {b^3} = \left( {a – b} \right)\left( {{a^2} + ab + {b^2}} \right)\)
5. \({a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} – ab + {b^2}} \right)\)

Q.5. Discuss the use of variables in finding the perimeter of a rectangle.
Ans: Let us consider a rectangle \(ABCD\), and the length of the side \(CD\), and breadth of the side \(BC\) is represented as \(l\) and \(b\).

Perimeter of a rectangle is the sum of the lengths of all sides.
\( = AB + BC + CD + AD\)
\( = l + b + l + b\)
\( = 2l + 2b\)
\( = 2\left( {l + b} \right)\)
Therefore, the perimeter of a rectangle \( = 2\left( {l + b} \right)\)

Summary

Variable is the value of an unknown quantity or number to be represented. They are represented using English alphabets such as \(x,y,p\) and \(t\). Variables are used in various common rules in different branches of mathematics, such as arithmetic, algebra and geometry. In geometry, formulate the perimeter and area of a rectangle and square. In arithmetic, we use the variables to represent the common rules such as commutative property, distributive property and associative property under various arithmetic operations. We use the variables to formulate the algebraic identities, expressions, and equations. Variables are also used to generalise the patterns and some mathematical sequences.

Frequently Asked Questions (FAQs)

Students might be having many questions regarding the Use of Variables in Common Rules. Here are a few commonly asked questions and answers.

Q.2. How do you define a variable in research?
Ans: A variable in research is simply a person, place, thing, or phenomenon that you are attempting to quantify in some way. The best way to realise the difference between a dependent and independent variable is to consider what the text tells us about the variable in question.

Q.3. Why is it important to use variables in Maths?
Ans: They make it possible to communicate mathematical ideas clearly and concisely. The equation \(2{x^2} + y = 6\) is much more straightforward than the equivalent phrase “two times some number times itself plus some other number equals six”. Variables also broaden the scope of mathematics.

Q.4. Why is it helpful to use a variable?
Ans: Variables are crucial to comprehend because they are the fundamental units of information studied and interpreted in research studies. In a descriptive study or an experiment, researchers carefully analyse and interpret each variable’s value(s) to make sense of how things relate to one another.

Q.5. What is the most important characteristic of a variable?
Ans: A variable is defined as any property, number, or quantity that can be measured or counted. A variable is also known as a data item. Variables include age, gender, business income and expenses, country of birth, capital expenditure, class grades, eye colour, and vehicle type.

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