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Algebra helps in solving linear equations with two variables.  Every linear equation with two unknowns has an unlimited number of solutions. To solve linear equation with 2 variables we consider the system of simultaneous linear equations (two linear equations in 2 variables). There are two methods for solving system of simultaneous linear equations. They are Substitution method and Elimination method.  By using Substitution method we substitute the value of the variable in the other equation. For elimination method multiply on or both equations by suitable numbers to transform them so that addition or subtraction will drop one variable.

The following examples help you to understand the methods of solving.

Example 1:- Use substitution method to solve for x in the system of equations.
10x + 3y= 10
3x + y = 2

Solution 1:- First, rewrite the second equation in the form of y

3x + y = 2

y = 2 – 3x

Now, substitute the second equation into the first equation

10x + 3y = 10

10 x + 3(2 – 3x) = 10

10x + 6 – 9x = 10 (combining like terms)

 x   = 4 

Substitute x= 4 in the equation y=   2 – 3x

 y =2 – 3(4)

 y = -10         
         
The value of x = 4 and y = -10. Verify your answer by substituting the x and y value in equation 1 or equation 2.

Example 2:-
Use elimination method to solve system of equations
6x – 3y = 6
12x + 3y = 30

Solution 2:- 

Add the first and second equation.

Hence we get,   18 x = 36 (divide by x on both sides)

x = 2

Now substitute x = 2 in the first equation

6 (2) – 3y = 6

12 – 3y = 6                 (use rules for solving)

y = -2

Therefore, the solution to the system of equation is x=2 and y = -2.
 

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