# Algebraic & Graphical interpretation of Simultaneous Equation

Let’s take a pair of linear equation:

……………..(1)

……………..(2)

With the help of coefficient of equations (1) & (2) we can easily determine if the solution of the simultaneous equations will be **Unique Solution, No Solution or Infinite Solution. **

Let’s concentrate on the coefficient of the equation now and calculate the following :

i.)

ii.)

iii.)

and compare the ratios.

**Unique solution : Algebraic Interpretation **

If **Number of solutions:** Unique solution **Graphical Representation : **Two intersecting lines **Type of Equation :** Independent**Example : **

2x – 3y = 1 ……………(1)

3x – 4y = 1 ……………(2)

In this case

So, . Hence Unique solution.

Unique solution : Graphical InterpretationLet's look at the graph of the two equations2x – 3y = 1 and 3x – 4y = 1on the same plot.

**No solution : Algebraic Interpretation **

If **Number of solutions:** No solution**Graphical Representation : **Two parallel lines**Type of Equation : **Inconsistent**Example : **

2x – 3y = 4 ……………(1)

4x – 6y = 12 ……………(2)

In this case

So, . Hence No solution.

No solution : Graphical InterpretationLet's look at the graph of the two equations2x – 3y = 4 and 4x – 6y = 12on the same plot.

**Infinite Number of solution : Algebraic Interpretation **

If **Number of solutions:** Infinite number of solution**Graphical Representation : **Same Line**Type of Equation : **Dependent**Example : **

2x – 3y = 4 ……………(1)

4x – 6y = 8 ……………(2)

In this case

So, . Hence, Infinite number of solution.

Infinite Number of solution : Graphical InterpretationLet's look at the graph of the two equations2x – 3y = 4 and 4x – 6y = 8on the same plot.

So, to solve a system of simultaneous equations, number of independent equations must be equal to the number of variables.