Elimination Method: Definition, Cases, and Calculations

The elimination method is one of two methods for solving the system of linear equations to evaluate unknown values. The linear equation is an equation that is written along with 2 variables, constant coefficients, and constants. Such as:

ax + by + c = 0

The above equation is a linear equation of two variables in which x and y are two unknown terms, “a” and “b” are constant coefficients and never be zero, and c is the constant term. When two linear equations are given, they form a system of linear equations.

What is the elimination method?

Elimination is a technique to find the unknown variables of the system of linear equations by solving them simultaneously. This method is used to create terms by multiplying with opposite coefficients for the sake of canceling one of the two variables. 

 The coefficients of the variables have the same or different signs (plus and minus). If the coefficients have the same sign, then we have to subtract the equations and if the coefficients have different signs (opposite signs) then we have to add the equations.

The addition or subtraction of the system of linear equations will eliminate one variable of the linear equation and we can evaluate the value of the second variable. After that, we can calculate the other variable by placing the value of the calculated variable in any equation. 

Cases of Elimination Method

There are two cases of elimination methods such as:

1. Eliminate the x variable
2. Eliminate the y variable

The above two are the ideal cases of the elimination method. You have to evaluate the equation for the variable x or y. take a look at the linear equation and make the same coefficients of those variables that have opposite signs.

1. If the coefficients of x are opposite, then by adding

ax + by = c

-ax + by = c

        2by = 2c

y = 2c/2b

y = c/b

2. If the coefficients of y are opposite, then by adding

ax + by = c

ax – by = c

 2ax      = 2c

y = 2c/2a

x = c/a

An elimination calculator by Allmath is a helpful way to solve the system of linear equations through the elimination method according to the above cases.

Steps to solve a system of linear equations using the elimination method

1. Make sure that the linear equations must be in a standard form such as ax + by = c and dx + ey = f
2. For solving linear equations through the method of elimination, one of two variables must have the same coefficients. If no variable has the same coefficients, then try any suitable number to multiply one equation or both equations to make the same coefficients of one variable.
3. On the basis of the sign between the similar variable, add or subtract the equations to eliminate x or y. 
4. After eliminating one variable, solve the equation for other variables to evaluate its value.
5. In the end, substitute the value of the evaluated variable to one of two equations and solve that equation for finding the value of an eliminated variable. 

Examples of elimination method 

Let’s take a few examples of solving the system of linear equations by the elimination method. 

Example 1

Solve the given system of linear equations using the elimination method to find the values of unknown variables. 

12x + 6y = 4

10x + y = 6

Solution

Step 1: Take the given linear equations and make the same coefficients of any variable. 

12x + 6y = 4

10x + y = 6

Multiply the second equation by 6.

6 * (10x + y) = 6 * 6

60x + 6y = 36

Step 2: Now solve the linear equations to eliminate “y”. Subtract the linear equations as coefficients of the variable “y” has similar signs.

   12x + 6y = 4

-(60x + 6y = 36)

     -48x + 0y = -32

-48y = -32

y = -48/-32

y = 24/16

y = 6/4

y = 3/2 = 1.5

Step 3: Now substitute the value of “y” to any equation to find the unknown value of the eliminated variable “x”.

12x + 6(3/2) = 4

12x + 3 (3) = 4

12x + 9 = 4

12x = 4 – 9 

12x = -5 

x = -5/12

The problems of the elimination method can also be done with the help of the elimination calculator by AllMath. Let us solve the above system of linear equations using the online calculator.

Example 2

Solve the given system of linear equations using the elimination method to find the values of unknown variables. 

p + 8q = 15

4p + 2q = 16

Solution

Step 1: Take the given linear equations and make the same coefficients of any variable. 

p + 8q = 15

4p + 2q = 16

Multiply the first equation by 4.

4 * (p + 8q) = 4 * 15

4p + 32q = 60

Step 2: Now solve the linear equations to eliminate “p”. Subtract the linear equations as coefficients of the variable “p” has similar signs.

   4p + 32q = 60

-(4p + 2q = 16)

   0p + 30q = 44

30q = 44

q = 44/30

q = 22/15

Step 3: Now substitute the value of “q” to any equation to find the unknown value of the eliminated variable “p”.

4p + 2 (22/15) = 16

4p + 44/15 = 16

4p = 16 – 44/15

4p = 16 – 2.94

4p = 13.06

p = 13.06/4

p = 3.265

Sum Up

Elimination is one of two methods for solving the system of linear equations. The other method is the substitution method. The elimination method eliminates one variable and evaluates the sending variable. This method is very easy to evaluate the unknown variables of the system of linear equations.