3.Simple Equations
Introduction to Simple Equations
• An equation is a condition of equality between two mathematical expressions.
Example: 2x − 3 = 5 , 3x + 9 = 11, 4y + 2 = 12
• If the left hand side of an equation is equal to its right hand side for any value of the variable,
then that value of variable is called the solution of that equation.
Example: For the equation, 5x + 5 = 15, x = 2 is a solution.
Solving Simple Linear Equation
A equation remains unchanged if
• The same number is added to each side of the equation.
• The same number is subtracted from each side of the equation.
Example: 5𝑥 + 3 = 13
On subtracting 2 from both sides, we get
5𝑥 + 3 − 2 = 13 − 2
5𝑥 + 1 = 11
• The same number is multiplied to each side of the equation.
• Each side of the equation is divided by the same number.
Example: 5𝑥 + 3 = 13
On subtracting 2 from both sides, we get
5𝑥 + 3 − 2 = 13 − 2
5𝑥 + 1 = 11
Short-Cut method(Solving an Equation by Transposing terms)
Any term of an equation may be shifted to the other side with a change in its sign without affecting the
equality. This process is called transposition.
So, by transposing a term
• We simply change its sign and carry it to the other side of the equation.
•
‘+‘ sign of the term changes to ‘—‘ sign to the other side and vice-versa.
•
‘×’ sign of the factor changes to ‘÷‘ sign to the other side and vice-versa.
•
Now, simplify L.H.S. such that each side contains just one term.
• Finally, simplify the equation to get the value of the variable.
For Example: To solve 2x + 8 = 24
Given, 2x + 8 = 24
Transposing 8 to the right hand side, we get
⇒ 2x = 24 − 8
⇒ 2x = 16
∴ x = 8
Hence, the value of x is 8.
Application of Simple Equations
There are several problems which involve relations among known and unknown numbers and can be
put in the form of equations. The equations are generally stated in words and it is for this reason we
refer to these problems as word problems. With the help of equations in one variable, we have already
practiced equations to solve some real life problems.
Steps involved in solving a linear equation word problem:
• Read the problem carefully and note what is given and what is required and what is given.
• Denote the unknown by the variables as x, y, …….
• Translate the problem to the language of mathematics or mathematical statements.
• Form the linear equation in one variable using the conditions given in the problems.
• Solve the equation for the unknown.
• Verify to be sure whether the answer satisfies the conditions of the problem.
An equation is a condition of equality between two mathematical expressions.
Eg:2x−3=5 , 3x+9=11, 4y+2=12
If the left hand side of an equation is equal to its right hand side for any value of the variable, then that
value is called the solution of that equation.
Eg:For the equation, 5x+5=15, x=2 is a solution.
When we add or subtract the same number to or from both the sides of an equation, the value of the
left hand side remains equal to its value on the right hand side.
Eg:5x+3=13
On subtracting 2 from both sides, we get
5x+3−2=13−2
5x+1=11
When we divide or multiply an equation on both the sides by a non-zero number, the value of the left
hand side remains equal to its value on the right hand side.
Eg: 1) 5x+1=13
On dividing both sides by 4, we get
2) 5x+1=13
On multiplying both sides by 4, we get
4(5x+1)=4(13)
20x+4=52
Application of Simple Equations
To find the solution of an equation, we have to perform identical mathematical operations on the
two sides of the equation so that only the variable remains on one side.
Eg:3x+8=84
3x+8-8=84−8
3x=76
Transposing means moving a term of the equation to the other side. Transposing a number is the
same as adding or subtracting the same number from both sides of the equation.
Eg:To solve 2x+8=24
Given, 2x+8=24
Transposing 8 to the right hand side, we get
⇒2x=24−8
⇒2x=16
x=8
Hence, the value of x is 8.
When a number is transposed from one side of the equation to the other, its sign changes
