# Solving Linear Equations with Two Variables

A first degree equation in two variables is known as a linear equation  in two variables. The equation contains two variables without exponents. Examples are:
$x-2y = 8$
$y = -3x + 5$
$C=2\pi r$
A single linear equation in two variables can be solved literal for one variable in terms of the other. A common solution for a system of linear equations consisting of two equations with the same two variables can be found if it exists using algebraic and graphing methods.

## Different forms of a linear equation

A linear equation with two variables is expressed as a straight line graph in coordinate plane and it can expressed in different forms.  The commonly used forms of a linear equation are,
1. General form                $ax+by+c=0$    Example: $2x-3y+5=0$
2. Standard form              $ax+by=c$         Example: $3x+5y=10$
3. Slope-Intercept form     $y=mx+b$          Example: $y=-2x+7$
4. Point-slope form     $y-y_{1}=m(x-x_{1})$     Example: $y-3=\frac{2}{3}(x-6)$
For the purpose of graphing the equation of a line given in any form is solved to write in slope intercept form. Slope intercept form is a literal equation solved for y in terms of x. x is called the independent variable and y the dependent variable.

## Solving a linear equation in slope intercept form

A linear equation in two variables is solved into slope intercept form using the properties of inequalities which are used to solve linear equations of one variable.
Example: Write the equation $x+2y=4$ in slope intercept form.
The equation is given in standard form. To write the equation in the standard form, we need to isolate the dependent variable y on one side of the equation.
$x+2y = 4$
$-x -x$                                          Subtraction Property of Equality
-------------------
$2y = -x + 4$
$\frac{2y}{2} =\frac{-x+4}{2}$                                   Division Property of Equality
$y=\frac{-x}{2}+2$                                Equation solved in slope intercept form.
Equations given in other forms can also be solved to slope intercept in a similar manner.

## Solving a system of Linear equations in two variables.

A system of linear equations will consist of two or more linear equations withe same variables. A system consisting of two equations are classified on the basis existence of a solution as
1. Consistent System:  The system may have a unique numeric solution or infinitely many solutions.
2. Inconsistent system: The system does not have a solution.  In this case, the graphs of two equations are parallel straight lines.
In the case of a consistent system, the system will have a unique solution when the two lines representing the equations in the system intersect at a point. The coordinates of the point of intersection gives the respective numerical solutions to the variables x and y. Such systems are called independent systems.

For a consistent system with many solutions the graphs of the two equations will be coincident, thus all the points on the straight line are solutions to the system. The system of linear equations is then known as a dependent system.