A *system of equations* is when we have two or more equations are working together.

In linear equation in one variable, only one variable is present whose value is to be found out and only one equation is sufficient to find the value of the variable.

A linear equation in two variables has two variables in the equation. Solving these equations is not possible with one equation only. To solve the **two** variables, we need **two** equations in two variables.

Let us see what such pairs look like algebraically.

The general form for a pair of linear equations in two variables x and y is

aand_{1}x + b_{1}y + c_{1}= 0a, where_{2}x + b_{2}y + c_{2}= 0aare all real numbers and_{1}, b_{1}, c_{1}, a_{2}, b_{2}, c_{2}a_{1}^{2}+ b_{1}^{2 }≠ 0, a_{2}^{2}+ b_{2}^{2 }≠ 0

Some examples of the pair of linear equations in two variables are:

2x + 3y – 7 = 0and9x – 2y + 8 = 05x = yand–7x + 2y + 3 = 0x + y = 7and17 = y

There are three possibilities:

- No solution
- One solution
- Infinitely many solutions

“Independent” means that each equation gives new information. Otherwise, they are “Dependent”.

Also called “Linear Independence” and “Linear Dependence”

Example:

x + y = 22x + 2y = 4

Those equations are “Dependent” because they are really the same equation, just multiplied by 2.

So, the second equation gave no new information.

Simultaneous Linear Equations are true at the same time. That means a certain solution, all equations are true.

Or we can say, a pair of values of *x* and *y* satisfying each of the equations in the given system of two simultaneous equations in x and y is called a **solution** of the system.

**Algebra vs Graphs**

We prefer to use algebra where the system of linear equations have more than 2 variables and can’t be solved by a simple graph.

**Solution of pairs of linear equations in two variables algebraically:**

**Solution by Substitution method:**

Let the pair of equations be *a _{1}x + b_{1}y + c_{1} = 0 *and

*a*.

_{2}x + b_{2}y + c_{2}= 0- From one of the equations, express one of the variables say
*y*in terms of the other variable i.e.,*x*. - Substitute the value of
*y*, obtained in above step, in other equation, the getting an equation in*x*. - Solve the equation and get the value of
*x*. - Substitute the value of
*x*in the expression for*y*obtained in the first step and get the value of*y*.

Solution by **Elimination method**, i.e., by **equating the coefficients**:

- In the two given equations, make the coefficients of one of the variables numerically equal. To do so, multiply these coefficients by suitable constant.
- Add or subtract the equations obtained in above step according to as the terms having same coefficients are of the opposite or of the same signs and get an equation in only one variable.
- Solve the equation found and get the value of one of the variable.
- Substitute the value of this variable in either of the two given equations and find the value of the other variable.

Solution by **Cross Multiplication method**:

Let the pair of equations be *a _{1}x + b_{1}y + c_{1} = 0 *and

*a*.

_{2}x + b_{2}y + c_{2}= 0To find the values of x and y, we have the formulae:

Now, you can use any of the methods mentioned above and solve the system of linear equations in two variables.