##### Chapter 11

## Algebra

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

#### Introduction

**algebra**. The study of algebra involves manipulating mathematical symbols, and deals with the study of their rules and application. In

**Algebra**, you must have come across the concepts of variables,

**expressions with variables,**and equations. Additionally, you also need to learn about multiple formulae and identities.

**Algebra **is the branch of mathematics that helps in deriving unknown quantities using equations and variables. In your life, you may have heard of some terms like bank interests, or percentage of profit. Did you know that you can use **algebra **to derive the true value of these unknown quantities, known as variables, by using something called an equation?

There are **more examples of variables **that you can find in your daily life. For example, the distance between your home and school is also a variable, and so are the prices of the grocery items that your parents buy at the nearby store. You can use algebraic equations and formulas to solve for these unknown variables.

Have you seen a letter being used to denote a number while talking about an unknown quantity? That letter is a variable. The value of a variable varies, hence the name. When used in an algebraic expression, the variable may denote different numbers, and maybe denoted by any alphabet. Conventionally, alphabets x, y, z, or a,b,c are the most widely used variables.

All algebraic expressions and terms must consist of at least one variable. The **idea of a variable** is the distinguishing factor between an algebraic and a numerical expression. You would be surprised to know that the presence of a variable in any algebraic expression expands the possibilities of determining the value of the expression to infinity.

A mathematical expression that is made up of variables, constants, and algebraic operations like addition, subtraction, etc. is called a mathematical expression or mathematical equation. Another important component of an algebraic expression is a term. A term is a representation of grouping together of more than one factor in an algebraic expression.

As you grow up and study more complex mathematical concepts, you will come across different types of algebraic expressions and equations.

**Linear Equations**

These are equations of the first order and have a single, homogeneous variable. They are used to represent a straight line on a coordinate system. It is represented by,

y= mx+b, where,

m is the slope of the line

b is the y-intercept

x and y are the coordinates of the x and y axes, respectively

**Non-linear Equations**

Polynomial equations of the second order in one variable. It is represented as a function of x as,

f(x)= ax2 + bx + c

**Solving Linear Equations**

I. Solve for x

x= 24+2x

Or, x-2x=24

Or, -(2-1)x=24

Or, -x= 24

Or, x = -24

II. Solve for x and y

x+y=15

x-2y=8

From the first equation,

x= 15-y

Substitute this in equation 2

(15-y)-2y= 8

Or, 15-y-2y= 8

Or, 15- (y+2y)=8

Or, 15-3y= 8

Or, 15-8= 3y

Or. 7=3y

Or 3y=7

Or, y= 7/3

Substituting this in equation 1,

x= 15-7/3

x= [3(15)-7]/3

x=[45-7]/3

x=38/3

x= 12^{2}/_{3}

Imagine that you are planning a holiday with your father and trying to understand important factors like time of travel, mode of transport, etc. to reach the destination. Here is some basic information that you have,

The bus that you are planning to take to Goa from Bombay travels at 60 kmph.

You also know that it takes 5 hours to reach your grandmother’s home from your place, and the former is 20km from Goa.

Can you find out the distance between Bombay and Goa using an algebraic expression?

Solution

Speed of the bus= 60 kmph

So, the distance (d) traveled in 5 hours can be represented by the equation

d= 60×5

d= 300

Therefore, the distance (D) from Bombay to Goa can be represented as

D= d+20

D= 300+20 (substituting the value of d)

D= 320 km

Did you know that you can get help from a simple everyday tool like a matchstick to learn **algebra? Matchstick patterns **are a great way to understand the **idea of a variable, **or **the use of variables in common rules.**

**Examples of matchstick patterns**

Using the following **matchstick patterns, **find out the rule to represent the number of matchsticks to be used in each pattern.

- Pattern of A
- Pattern of T

Solutions

To write the letter A, you need 3 matchsticks. Therefore, the rule to represent this is 3n, where n is the number of matchsticks.

To write the letter T, you need 2 matchsticks. Therefore, the rule to represent this is 2n, where n is the number of matchsticks.

You can find **more matchstick patterns **to learn how to come up with algebraic rules using variables and constants.

**What are the basics of algebra?**

The basics of algebra consist of variables, constants, and mathematical operations like addition, subtraction, etc.

**How do you calculate algebra?**

You can calculate algebra with the use of algebraic expressions and equations.

**Who invented algebra?**

Al-Khwarizmi is credited with the invention of Algebra.

**Why is algebra so hard?**

The use of symbols and equations may make algebraic expressions look complicated.

**What does XY mean in algebra?**

XY is used to denote a term or the product of two variables X and Y.

**What are the four rules of algebra?**

The four basic rules of algebra include

- Symmetry rule
- Commutative rule
- Inverse of adding
- Two rules of equation

**Algebra **can be fun too, if you want it. With the MSVGo app, you can now have access to an interactive learning platform and vast video library, where you can learn about the most important algebraic modules from experts and peers. You can simply download the app to start learning today!