Linear Equations in two Variables
Linear equation in two variables x and y is an equation of the form ax + by + c = 0, where a, b and c are real numbers and both a and b are not equal to 0 at a time.
Examples of linear equation in two variables are:
2x +
3y = 6, x + 4y = 7
These equation can also written in the form
2x +
3y – 6 = 0, x + 4y – 7 = 0
respectively.
Example 1: Write each of the following equation in the form of
ax + by + c = 0 and indicate the values of a, b and c in each case:
(i) 3x + y = 7
(ii) 2y – 4 = x
Solution:
(i) 3x + y = 7 can be written as 3x + y – 7 = 0
Here, a = 3, b = 1 and c = 7
(ii) 2y – 4 = x can be written as 2y – 4 – x
= 0 or x + 2y – 4 = 0
Here,
a = 1, b = 2 and c = 4
Note: Equations of the type ax + b = 0 are also
examples of linear equations in two variables because they can be expressed as
ax + 0.y + b = 0. For example 4  5x = 0 can be written as 5x + 0.y + 4 = 0
Solving Linear Equations
Solving linear a equation ax + by + c = 0 is to find
the points (x, y) or pair of values of x and y which satisfy the given linear
equation. For example the point (2, 3) is the solution of the linear equation
2x + 3y – 13 = 0. Because for the value of x = 2 and y = 3 satisfy the equation
2x + 3y – 13 = 0,
i.e 2x2 + 3x3 – 13 = 0
or, 4
+ 9 – 13 = 0
or, 13
– 13 = 0
or, 0
= 0 (True)
Note:
 A
linear equation has infinitely many solutions.
 Linear
equation always represent a straight line joining those points in graph.
Example 2: Find any two solutions of the given linear equations:
(i) 3x+4y=10
(ii) y4=3x
Solution:
(i)
Here,
3x
+ 4y = 10
or,
3x = 10 – 4y
or,
x = (10 – 4y)/3
x

2

2

y

1

4

∴ The two solution
are (2, 1) and (2, 4)
(ii)
Here,
y
– 4 = 3x
or,
y = 3x + 4
x

1

1

y

7

1

∴ The two solution
are (1, 7) and (1, 1)
Graphing Linear Equations
A linear equation represents a straight line in the
graph. Graphing linear equation is
the plotting of the solution points of the equation and joining straight line
in the graph.
Some
of special cases:
(i) x=0 is the equation of yaxis and y=0
is the equation of the xaxis.
(ii) The graph of x=a is a straight line
parallel to yaxis.
(iii) The graph of y=b is a straight line
parallel to xaxis.
(iv) The graph of the equation of the form
y=mx is a line which always passes through the origin.
Example 3: Draw the graph of x + y = 7.
Solution:
Here,
x + y = 7
or, x = 7 – y
x

4

3

y

3

4

∴ The two solution of the given equation are (4, 3) and (3,
4). Now, plotting these two points in a graph, and joining a straight line
we get as given below,
Example 4: Draw the graph of 2x – y = 1.
Solution:
Here,
2x – y = 1
or, 2x = 1 + y
or, x = (1 + y)/2
x

1

2

y

1

3

∴ The two solution of the given equation are (1, 1) and (2,
3). Now, plotting these two points in a graph, and joining a straight line
we get as given below,
Example 5: Draw the graph of 2x + 3y = 6.
Solution:
Here,
2x + 3y = 6
or, 2x = 6 – 3y
or, x = (6 – 3y)/2
x

3

0

y

0

2

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