**Nature of Solutions of Linear Equations in Two Variables :**

In this section, we will learn how to find the nature of solutions of linear equations in two variables.

By solving any linear equation in two variables, we may have the following solutions.

(i) Unique solution

(ii) Infinitely many solution

(iii) No solution.

To apply the concept given below, the given equations will be in the form

a_{1}x + b_{1}y + c_{1} = 0

a_{2}x + b_{2}y + c_{2} = 0

(i) a_{1}/a_{2} ≠ b_{1}/b_{2, }we get a unique solution

(ii) a_{1}/a_{2} = a_{1}/a_{2 } = c_{1}/c_{2}, there are infinitely many solutions.

(iii) a_{1}/a_{2} = a_{1}/a_{2 } ≠ c_{1}/c_{2}, there is no solution

**Example 1 :**

On comparing the ratios a₁/a₂, b₁/b₂ and c₁/c₂, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident.

(i) 5 x – 4 y + 8 = 0

7 x + 6 y – 9 = 0

**Solution :**

From the given equations, let us find the values of a_{1}, a_{2},
b_{1}, b_{2}, c_{1} and c_{2}

a_{1} = 5, b_{1 } = -4, c_{1 }= 8

a_{2} = 7, b_{2 } = 6, c_{2 }= -9

a_{1}/a_{2} = 5/7 ------(1)

b_{1}/b_{2 } = -4/6 ------(2)

c_{1}/c_{2 } = -8/9 ------(3)

(1) ≠ (2)

Here, a_{1}/a_{2} ≠ b_{1}/b_{2}

Hence it has unique solution.

(ii) 9 x + 3 y + 12 = 0

18 x + 6 y + 24 = 0

**Solution :**

From the given equations, let us find the values of a_{1}, a_{2}, b_{1}, b_{2}, c_{1} and c_{2}

a_{1} = 9, b_{1 } = 3, c_{1 }= 12

a_{2} = 18, b_{2 } = 6, c_{2 }= 24

a_{1}/a_{2 } = 9/18 = 1/2 ------(1)

b_{1}/b_{2 }= 3/6 = 1/2 ------(2)

c_{1}/c_{2 }= 12/24 = 1/2 ------(3)

(1) = (2) = (3)

Here a_{1}/a_{2} = b_{1}/b_{2 }= c_{1}/c_{2}

The given lines are having infinitely many solution.

(iii) 6 x - 3 y + 10 = 0

2 x - y + 9 = 0

**Solution :**

From the given equations, let us find the values of a_{1}, a_{2}, b_{1}, b_{2}, c_{1} and c_{2}

a_{1} = 6, b_{1 } = -3, c_{1 }= 10

a_{2} = 2, b_{2 } = -1, c_{2 }= 9

a_{1}/a_{2} = 6/2 = 3

b_{1}/b_{2 } = -3/-1 = 3

c_{1}/c_{2 } = 10/9

Here, a_{1}/a_{2 } = b_{1}/b_{2} ≠ c_{1}/c_{2}

Hence it has no solution.

After having gone through the stuff given above, we hope that the students would have understood, nature of solutions of linear equations in two variables.

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