Equations with Variables

An equation in algebra consist of two algebraic expressions separated by an inequality sign generally equal to. For example, ax + b = c. These equations consist of some numbers, a variable, operator sign and inequality sign. An equation with variable is used to represent a general condition . For example, if we ask, if three more than twice of a number is equal to five, then what is the number. In this case, we actually don’t know the answer however we can represent it as 2x + 3 = 5, here x is the variable which represent the unknown number.

In this article, we will read in detail about equation with variables, types of variables and equation with variables on both sides and solve problems based on them.

What is an Equation?

An equation is a mathematical statement that proves the equality of two expressions. It is made up of two sides: the left-hand side (LHS) and the right-hand side (RHS), which are divided by an equals sign. The equals symbol signifies that the expressions on both sides are equal in value.

For Example, 3x + 2 = 5

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 value.

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 is Variable
• 2 is coefficient of x
• 8 and 20 is constants
• ‘-’ is operator

Types of Variables

There are two types of variables, namely

• Dependent Variable
• Independent Variable

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.

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…

Equations with Variables on Both Sides

Equations with variables on both sides are those in which variables occur on both sides of the equal sign. To solve these equations, move terms from one side to the other in order to isolate the variable and determine its value.

How to Solve Equation with Variable on Both Sides?

Follow the below mentioned steps to solve equations with variable on both sides:

Step 1: Simplify each side of the problem by grouping like terms.

Step 2: Isolate a variable on one side of an equation, shift the variable-containing terms from one to the other.

Step 3: After relocating terms and isolating the variable on one side of the problem, use additional simplification as needed.

We can understand it better through the examples solved below.

Also, Read

• Linear Equations in One Variable
• Difference Between Constant and Variables
• Pair Of Linear Equations In Two Variables

Solved Examples on Equations with Variables

Example 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 

Example 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

Example 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

Example 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 

Example 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

Example 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

Example 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 

x + y  = 6 + 23 = 29 marbles