Equations with Variables

Algebra is one of the oldest branch of mathematics that deals with number theory, geometry, and analysis is termed as Algebra.  According to some definitions. it is defined as the analysis of mathematical symbols and laws, as well as the manipulation of these mathematical symbols.

Algebra also includes a wide range of topics, from solving basic problems to studying abstractions. basically focused on variable which has undefined values .The branch of mathematics that deals with variables  and the arithmetic operations across these symbols. These variables do not have any fixed values . In our day to day life problem. There are certain values that keep on changing.  

Here in algebra, these values or variables are often represented by such as x, y, z, p, or q. These variables are manipulated through various arithmetic operations of addition, subtraction, multiplication, and division, with an objective to find the values.

Expressions

A combination of terms that are combined by using mathematical operations such as subtraction, addition, multiplication, and division is termed Expression.

• A constant is a fixed numerical value.
• A variable is a symbol that does not have a fixed value.
• A constant, a single variable, or a combination of a variable and a constant combined with multiplication or division is defined as a Term.

Example of Expression in Algebra : 3x + 9 , 5x + 10 …

Variables in Equations

Variable is defined as the alphabetic character that expresses a numerical value or a number. In algebra, a variable is used to represent an unknown quantity or unknown values.

These alphabets most commonly, ‘a’,’b’,’c’, ‘x’,’y’ and ‘z’ are used as variables in equations. Variable is a term that defines the value of another variable or equation 

For example, in equation, 2x – 8 = 20

x → Variable

2 → coefficient of x

8 and 20→ constants

‘-’ → operator

Types of Variables

There are two types of variables, namely

• Dependent Variable: The variable whose value depends on the estimation of another variable in its condition is termed a dependent variable …

For example: Consider the condition y = 5x + 3. In this equation, the value of the variable ‘y’ changes as per the adjustments in the value of ‘x’. 

• Independent Variable: An independent variable is a variable that describes a variable whose values are independent of changes. Suppose If x and y are two variables in an algebraic equation then every value of x is linked with any other value of y. Here  value of ‘y’ is said to be a function of x value known as an independent variable, and  value of ‘y’ is known as a dependent variable.

For example: In this function y = x², here x is an independent variable and y is a dependent variable whose value is dependent on the value of x.

Some examples of the variable equation or algebraic expression:

• y = 5x +10  
• 3y = 99x + 9
• 3x + 9 
• 3x² + 5x 

In this way, a variable is used in an equation to find the other variable or to complete the equation…

Sample Problems

Problem 1: Find the value of x: 5(x + 2) = 3x – 8

Solution:

We have 

5(x + 2) = 3x – 8                             {here x is independent variable} 

5x + 10 = 3x – 8 

by adding or subtracting like terms together 

5x – 3x = – 8 + 10 

       2x =  2 

         x = 1

here the value of x is 1 

Problem 2: Simplify the equation 3x + 2y = 16 : if x = 2, find y?

Solution:

Given equation : 3x + 2y = 16

                               here x = 2 

In above equation y is dependent variable whose value is dependent on the value of x 

put the value of x in equation 

3 (2) + 2y = 16 

6 + 2y = 16 

       2y = 16 – 6

       2y = 10 

         y = 10/2 

         y = 5

Therefore the value of y is 5

Problem 3: Simplify equation  : 2x + 5y² = 40 and find value of x, if y = 2?

Solution:

Given equation : 2x + 5y² = 40

y = 2 

in this equation x is dependent variable whose value is depend on the value of y . whereas y is independent

we have 

2x + 5y² = 40

2x + 5 (2)² = 40 

2x + 5 (4) = 40

    2x + 20 = 40 

            2x = 40 – 20 

            2x = 20 

             x  = 10

hence the value of x is 10

Problem 4: There are 30 pens in a bag. Write the variable expression (algebraic expression) for the number of pens in x number of bags.

Solution:

The number of pens in one bag =  25

The number of bags =  x

So the number of pens in  x bags =  25x 

here 25x is the variable expression 

lets assume numbers of bags x = 2

then the solution will be 25x = 25 × 2 = 50 pens 

Problem 5: Solve the equation 5x – 10 = 3x – 8.

Solution:

Given,       5x – 10 = 3x – 8

 Adding 10 on both sides,

         5x – 10 + 10 = 3x – 8 + 10

                         5x = 3x + 2

Subtract 3x from both sides,

              5x – 3x = 3x + 2 – 3x

                      2x = 2

Dividing both sides of the equation by 2,

                   2x/2 = 2/2

                         x = 1

Problem 6: Evaluate the given variable expression for a = 7; b = −3 and c = 2, 3ab+7bc+9ca.

Solution:

The given algebraic expression is  3ab+7bc+9ca

Substitute the below values in the above expression:

a=7;b=−3;c=2

3ab+7bc+9ca

=3(7)(−3)+7(−3)(2)+9(2)(7)

=−63−42+126

=−105 + 126

= 21

Problem 7: Arun has 6 marbles. Annie has 5 more than 3 times the number of marbles Arun has. Write the expression and Find the total number of marbles that Arun and Annie have together?

Solution:

Here given Arun marbles (x) = 6 

lets assume the marbles annie has (y)   

As per the question : Annie has 5 more than 3 times the number of marbles Arun has

Therefore,           y = 5 + 3x        is an expression 

                             = 5 + 3(6)               { Arun marbles (x) = 6 }

                             =  5 + 18 

                          y = 23 

hence the number of marbles annie has y = 23 

and total number of marbles = x + y 

                                              = 6 + 23

                                              = 29 marbles