Linear Equation in Two Variables

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Definition of Linear equation

A linear equation in two variables is defined as

ax + by = r

Where a and b are the coefficients of linear equation and r is the constant. Any of the constants (a, b and c) can be zero but not all. This equation has two variables x and y, both having degree of 1. Note that for the equation to be linear, both of the variables’ degree should not be greater than 1 and at least one of the variables’ degree is 1. 

Example

3x + 2y = 6 → This is an example of linear equation in variables x and y.

x2 + 3y = 6 → This is not an example of linear equation in two variables because the degree ( highest power ) of the equation is 2.

xy = 5 → This is not an example of linear equation in two variables because the equation involves the product of variables x and y.

Solution of linear equation

The solution of the linear equation represents a point (u, v), when inserted in place of variables x and y satisfies the equation i.e.

ax + by = r

if u and v is the solution of the equation. Then,

a(u) + b(v) = r
a(u) + b(v) – r = 0

The above equation is satisfied.

Example

The point (4, -3) is a solution of the equation because it satisfies the given equation i.e.                                    

The equation is satisfied.

The point (2, 1) is not the solution of the equation because it does not satisfy the given equation i.e.

3x + 2y = 6
3(2) + 2(1) = 6
8 – 6 = 0
2 ≠ 0

The equation is not satisfied.

Graphical Representation of a linear equation

The linear equation  represents a straight line in two dimensional system.

ax + by = r
by = -ax + r
dividing by b on both sides
y = (-a/b)x + (r/b)

-a/b is the slope of the line  i.e. shows the direction and steepness of the line.

r/b is the y-intercept of the line i.e. y-intercept is a point where the line intercepts the y-axis.

The solution of the linear equation represents a point in two dimension, which lies on the line. In the above example, the point (4, -3) lies on the line . However, the point (2, 1) does not lie on the line.

Let’s take one more equation and draw its graph to clearly understand the slope and Y-intercept.

Equation: 2x + 3y = 6
Linear Equation of a line with intercept and slope

Different Examples of Linear equation

Consider the same equation ax + by = r

  • If the coefficient a = 0. Then, the equation becomes or y = r/b. This represents a horizontal line parallel to the x-axis at a distance of r/b from it with the slope of 0 and y intercept of r/b.
  • If the coefficient b=0. Then the equation becomes or ax =r or x = r/a. This represents a vertical line parallel to the y-axis at a distance of r/a from it with the undefined slope and no y-intercept (the line never intercepts the y-axis)

System of Linear Equations in two variables.

The system of linear equations in two variables consists of two or more linear equations in which all the equations of the system are considered simultaneously. i.e.

ax + by = r
cx + dy = s

represents a system of two linear equations in two variables.

Solution
A solution of a system of linear equation consists of point(s) that satisfies each of the linear equation in the system. Graphically, it represents point(s) at which all of the lines in the system intersect.

Types of System

There are two types of system depending upon the solution obtained.

Consistent System: A system of linear equations is said to be consistent if the system has at least one solution.

Example

3x + 2y = 6 → equation 1 
 x + y = 3 → equation 2

These two equations represents a consistent system because the solution of the system exists at point (0,3).

Inconsistent System: A system of linear equations is said to be inconsistent if the system has no solution.

Example

3x + 2y = 6 → equation 1 
6x + 4y = 5 → equation 2

They represents an inconsistent system because the solution of the system does not exist. Both lines are parallel to each other.

Types of Equation in the System

There are two types of equation in the system depending upon the solution obtained.

Dependent Equations: The equations in the system are said to be dependent if all the solutions of all the equations in the system are the same. Graphically, all the lines in the system are on the top of one another.

Example

3x + 2y = 6 → equation 1  
9x + 6y = 18 → equation 2

The above system consists of dependent equations because both of the lines are identical. If we multiply equation 1 by 3 we get the exactly 2nd equation that’s why they are not different to each other. 

Independent Equations: The equations in the system are said to be independent if the solution consist of only one point.

Example 

3x + 2y = 6 → equation 1  
x + 4y = 3 → equation 2

The above system consists of independent equations because the system has only one solution i.e. the lines intersect at exactly one point.

Note that if the system is consistent and equations are not independent then all the lines in the system will intersect at only one point.

Finding the Solution of System of Linear equations

The solution of the system of linear equations can be found by one of the following three methods. For simplicity, we will consider the system of two linear equations.

  1. Graphical Method
  2. Substitution Method
  3. Elimination Method

Graphical Method: The solution of the system through this method can be obtained by plotting each of the lines in the 2 dimension and then finding a point at which all of the lines in the system intersect.

Substitution Method: The solution of the system through this method can be found by solving one of the two equations for one variable and substituting the value of that variable in the other equation. This method is useful when the solution involves integer value. However, it gets tedious when fractions or decimals are involved.

Example

Consider the following system of linear equation

3x + 2y = 6 → equation 1  
x + y = 3 → equation 2

Taking the 2nd equation and solving it for the variable y

x + y = 3

y = 3 – x → substituting this value of y in the equation 1

3x + 2( 3 – x )  = 6
3x + 6 – 2x = 6
3x – 2x = 6 – 6
x = 0

To get the value of y, the value of the x can be substituted in any of the above equations. Let’s substitute it in equation 2.

x + y = 3
y = 3 – x
y = 3 – 0
y = 3

Hence, the solution of the system is (0,3).

Elimination Method: The solution through this method is found by eliminating one of the variables in the system of equations. This method often works by multiplying the coefficients by such numbers so that one of the variables of both equations have same number but opposite signs such that when the equations are added, that variable is eliminated. This method is also known as Addition Method.

Example

Consider the same set of linear equations

3x + 2y = 6 → equation 1  
x + y = 3 → equation 2

Multiplying the ‘equation 2’ by -2 to eliminate the variable x i.e.

-2x – 2y = 6 

Adding that equation to that equation 1, we get

x = 0

The value of the x can be substituted in any of the above equations to get the value of y. Let’s substitute  it in equation 2. We get

0 + y = 3 
y = 3

Hence, the solution of the system is (0,3).

 

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