# Pair Of Linear Equation In Two Variables

An equation is an expression with equality sign on both the sides. A polynomial involves a mathematical expression with powers of the variables as non-negative integers.

For example, $x^4~+~3x^3~+~2x^9$ is a polynomial but $x^{\frac 35}~+~3x^{0.6}$  is not a polynomial.

While defining polynomials we should know about the concept of ‘degree’. Degree can be defined as the highest power of the variable in the given polynomial.

A polynomial with degree 1 is called a linear polynomial. A polynomial with degree 2 is called a quadratic polynomial and a polynomial with degree 3 is called cubic polynomial.

In the upcoming discussion, we will learn about the concept of polynomials with degree 1.

Let us start with the practical application of simultaneous linear equations in our day to day lives. Pair of linear equations can find its way in every possible scenario. Say you went to fish market to buy fishes. There were two sizes of fishes available. The fisherman told that the total price of the smaller fish is 3 times the total price of the bigger fish. Also, the total money that you bought from your home is Rs.100. Can you find out how much did you spent to buy the two types of fishes?

Let us understand this mathematically,

Let the price of the smaller fish be Rs.x and the price of the bigger fish be Rs.y.

As per the first condition, x = 3y…………(1)

And as per the second condition, x + y = 100……………(2)

In order to find the solution, we need to solve both the equations and find the value of both x and y.

In earlier classes we have studied about linear equation in one variable and we know how to solve it. If there was one variable and one equation, we used to solve it easily, but in this case we have two variables and two equations.

One thing is for certain that, we need two different set of linear equations in order to find out the two different unknowns. If one equation is given and two variables are asked to be solved, we will not a particular solution.

For example,

3x + 2y = 9 and 5x + y = 10

These simultaneous equations can be solved and we can arrive at a particular solution from these, but on the other hand,

6x + 7y = 9

Here, we cannot get a particular solution for this as there is only one condition given and we have two unknowns. We can rewrite above equation as:

y = $\frac{9~-~6x}{7}$<

Depending on the values of x, the values of y will change accordingly. So, one unique solution is not possible.

Thus, it can be clearly said that in order to get a particular solution of systems of linear equations in two variables, we need two different sets of independent conditions.