A Linear Equation in one variable is defined as ax + b = 0 or ax = c, where a, b and c are real numbers. Also, a ≠ 0 and x is an unknown variable.

The solution of the equation ax + b = 0 is x = - b/a. We can also say that - b/a is the root of the linear equation ax + b = 0.

A Linear Equation in two variables is of the ax + by + c = 0 or ax + by= d type where a, b, c and d are constants and also, both a and b are not equal to 0.

**Substitution Method**Step 1: Find the value of one variable say y in terms of the other i.e. x. from either equation.Step 2: Then substitute the value of y so obtained in the other equation. Therefore, we have a single equation in one variable x.Step 3: Now solve this equation for x.Step 4: In the end, substitute the value of x, thus obtained, in first step and find the value of y.**Method of Elimination**

Step 1: Multiply both the equations with such numbers so as to make the coefficients of one of the two unknowns numerically same.Step 2: To get an equation containing only one know n, subtract or add the two equations. Solve this equation to get the value of the unknown.Step 3: In either of the two original equations, substitute the value of the unknown. Thus, by solving that, the value of the other unknown is obtained.**Short – Cut Method**

Let us consider two equations as: a_{1}x + b_{1}y = c_{1}and a_{2}x + b_{2}y = c_{2}Then, the solution will be written as x/(b_{1}c_{2}b_{2}c_{2}) = y/(c_{1}a_{2}c_{2}a_{1}) = (-1)/(a_{1}a_{2}a_{2}b_{1})i.e. x = - (b_{1}c_{2}- b_{2}c_{1})/(b_{1}c_{2}b_{2}c_{1}) and y = (c_{1}a_{2}- c_{2}a_{1})/(a_{1}b_{2}- a_{2}b_{1})

Suppose, we have two linear equations: a_{1}x + b_{1}y = c_{1} and a_{2}x + b_{2}y = c_{2}

Then

- If a
_{1}/a_{2}= b_{1}/b_{2}, the system will have only one solution that will be consistent. The graphs of this type equation will have intersecting lines. - If a
_{1}/a_{2}= b_{1}/b_{2}= c_{1}/c_{2}, the system will be consistent with numerous solutions. The graphs of this type of equation will have coincident lines. - If a
_{1}/a_{2}= b_{1}/b_{2}≠ c_{1}/c_{2}, the system will have no solution and will be consistent. The graphs of this type of equation will have parallel lines.