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) y-4=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 y-axis and y=0 is the equation of the x-axis.

(ii) The graph of x=a is a straight line parallel to y-axis.

(iii) The graph of y=b is a straight line parallel to x-axis.

(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

∴ The two solution of the given equation are (3, 0) and (0, 2). Now, plotting these two points in a graph, and joining a straight line we get as given below,