The document discusses linear expressions and equations. It provides examples of linear and nonlinear expressions, defines equations as statements of equality involving variables, and describes methods for solving different types of linear equations. These include balancing both sides of an equation, transposing terms between sides, solving equations with variables on one or both sides, and reducing equations to simpler or linear forms. Applications of linear equations to problems involving numbers, ages, and finances are also covered.

1. Pooja K More

2. Some examples of expressions
• 5x
• 2x – 3
• 3x + y
• 2xy + 5
• xyz + x + y + z
• x2 + 1
• y + y2

3. Some examples of equations
• 5x = 25
• 2x – 3 = 9
• 2y +
5
2
= 8
• 6z + 10 = -7
• 9x – 11 = 8

4. equations use the equality (=) sign
If yes = equation 
If no = equation 
• For example,
2xy + 5 has two variables.
2.1 Introduction

5. These are linear expressions:
• 2x
• 2x + 1
• 3y – 7
• 12 – 5z
These are not linear expressions
• 𝑥2
+ 1
• y+𝑦2
• 1+z+𝑧2
+𝑧3
(since highest
power of
variable > 1)

6. • An algebraic equation is an
equality involving variables.
• It has an equality sign.
• The expression on the left of
the equality sign is the Left
Hand Side (LHS).
• The expression on the right
of the equality sign is the
Right Hand Side (RHS).

7. • In an equation the values of the expressions on the
LHS and RHS are equal.
• This happens to be true only for certain values of the
variable.
• These values are the solutions of the equation.
• x = 5 is the solution of the equation 2x – 3 = 7.
For x = 5, LHS = 2 × 5 – 3 = 7 = RHS
• On the other hand x = 10 is not a solution of the
equation.
• For x = 10, LHS = 2 × 10 –3 = 17. This is not equal to
the RHS

8. • We assume that the two
sides of the equation are
balanced.
• We perform the same
mathematical operations on
both sides of the equation,
so that the balance is not
disturbed.
How to find the solution of an equation?

9. Algebraic Expressions
Any expression involving
constant, variable and some
operations like addition,
multiplication etc is
called Algebraic Expression.

10. •A variable is an unknown number
and generally, it is represented by
a letter like x, y, n etc.
•Any number without any variable
is called Constant.
•A number followed by a variable
is called Coefficient of that
variable.
•A term is any number or variable
separated by operators.

11. Equation
A statement which
says that the two
expressions are
equal is
called Equation.

12. Linear Expression
A linear expression is an expression whose highest
power of the variable is one only.

13. Linear Equations
The equation of a straight line is the linear equation. It could be in one
variable or two variables.
Linear Equation in One Variable
If there is only one variable in the equation then it is called a linear
equation in one variable.
The general form is
ax + b = c, where a, b and c are real numbers and a ≠ 0.

14. Example
x + 5 = 10
y – 3 = 19
These are called linear equations in one variable
because the highest degree of the variable is one and
there is only one variable.

15. Some Important points related to Linear
Equations
• There is an equality sign in the linear equation. The
expression on the left of the equal sign is called the
LHS (left-hand side) and the expression on the right
of the equal sign is called the RHS (right-hand side).
• In the linear equation, the LHS is equal to RHS but
this happens for some values only and these
values are the solution of these linear equations.

18. 2.2 Solving Equations which have Linear
Expressions on one Side and Numbers on the
other Side
There are two methods to solve such type of
problems-
1. Balancing Method
2. Transposing Method

19. 1. Balancing Method
• In this method, we have to add or subtract
with the same number on both the sides
without disturbing the balance to find the
solution.
Example
Find the solution for 3x – 10 = 14

20. Step 2: Now to balance the equation, we
need to divide by 3 into both the sides.
3𝑥
3
=
24
3
x = 8
Hence x = 8 is the solution of the equation.

21. We can recheck our answer by substituting the value of x
in the equation.
Here, LHS = RHS, so our solution is correct

22. 2. Transposing Method
• In this method, we need to transpose or
transfer the constants or variables from one
side to another until we get the solution.
• When we transpose the terms the sign will get
changed.
Example
Find the solution for 2z +10 = 4.

23. Solution
Step 1: We transpose 10 from LHS to RHS so that all
the constants come in the same side.
2z = 4 -10 (sign will get changed)
2z = -6

24. 2𝑧
2
=
−6
2

25. 2.3 Some Applications of Linear Equation
We can use the concept of linear equations in our daily
routine also. There are some situations where we need
to use the variable to find the solution. Like,
•What number should be added to 23 to get 75?
•If the sum of two numbers is 100 and one of the
no. is 63 then what will be the other number?

26. We begin with a simple example.
Sum of two numbers is 74.
One of the numbers is 10 more than the other.
What are the numbers?
We have a puzzle here.
We do not know either of the two numbers, and
we have to find them.

27. We are given two conditions.
(i) One of the numbers is 10 more than the
other.
(ii) Their sum is 74.
If the smaller number is taken to be x, the larger
number is 10 more than x, i.e., x + 10.
The other condition says that the sum of these two
numbers x and x + 10 is 74.

28. This means that x + (x + 10) = 74.
or 2x + 10 = 74 Transposing 10 to RHS,
2x = 74 – 10 or 2x = 64
Dividing both sides by 2, x = 32.
This is one number.
The other number is x + 10 = 32 + 10 = 42
The desired numbers are 32 and 42.
(Their sum is indeed 74 as given and also one number is 10
more than the other.)
We shall now consider several examples to show how useful
this method is.

37. 2.4 Solving Equations having the Variable
on both Sides
• An equation is the equality of the values of two expressions.
• In the equation 2x – 3 = 7, the two expressions are 2x – 3
and 7.
• In most examples that we have come across so far, the RHS
is just a number.
• But this need not always be so; both sides could have
expressions with variables.

38. For example, the equation
2x – 3 = x + 2
has expressions with a variable on both sides;
the expression on the LHS is (2x – 3)
and
the expression on the RHS is (x + 2).

46. 2.5 Some More Applications

50. 2.6 Reducing Equations to Simpler Form

53. 2.7 Equations Reducible to the Linear
Form
Sometimes there are some equations which are not
linear equations but can be reduced to the linear form
and then can be solved by the above methods.

56. WHAT HAVE WE DISCUSSED?
• An algebraic equation is an equality involving variables. It says
that the value of the expression on one side of the equality
sign is equal to the value of the expression on the other side.
• Just as numbers, variables can, also, be transposed from one
side of the equation to the other.
• The utility of linear equations is in their diverse applications;
different problems on numbers, ages, perimeters, combination
of currency notes, and so on can be solved using linear
equations.