Elimination Method

The elimination method is a technique used to solve linear equations by eliminating a variable, either by multiplying or dividing both equations to make the coefficients of one variable equal. Then, using basic arithmetic operations to eliminate a variable; hence, the process is also called the addition-subtraction method.

When solving linear equations using the elimination method, we remove one variable and then solve the equation using addition or subtraction. Follow these steps to use the elimination method:

Step 1: Write the equations in standard form

Before solving the equations, check if they are in standard form. If not, arrange them in the form ax + by = c.

 
Step 2: Multiplying or dividing the equation

First, we multiply or divide one or both equations by a non-zero number to make the coefficients of a variable equal. 

Step 3: Adding or subtracting the equation to eliminate a variable 

Since the coefficients of a variable in both equations are the same, we can add or subtract the equations to eliminate that variable.

Step 4: Simplifying the equation

Now we have an equation with one variable, and by solving the equation, we can find the value of this variable. 

Step 5: Substitute to find the other variable

To find the value of the other variable, we substitute the value we found in the given equation and solve the equation. 

For example, solving 2x + 3y = 12 and 4x - 3y = 6

• Here, both equations are of the form ax + by = c and have one variable that has a common coefficient.
• That is the coefficients of y are 3 and -3, as they have opposite signs, we add the equations.

1. Adding the equations:
   
   \((2x + 3y) + (4x - 3y) = 12 + 6 \\ 6x + 0y = 18 → 6x = 18\\ x = 18/6 \\ x = 3\\\)
   So, the value of x = 3
    
2. Finding the value of y by substituting the value of x in 2x + 3y = 12
   
   \(2(3) + 3y = 12 \\ 6 + 3y = 12\\ 3y = 12 - 6 \\ 3y = 6\\ y = 6/3 = 2\\\)
   Therefore, x = 3 and y = 2

When the equations are coincident lines, which means both equations are on the same line. Then there will be more than one point of intersection, that is, every point on the line satisfies both equations.
 

For these equations, if we use the elimination method, the answer will be 0 = 0. That means there are infinitely many solutions, as x and y cancel out.

Let take an example of linear equation in one variable.
 

• Equation 1 → 2(x + 3) - 5 = 0
• Equation 2 → 2x + 1 = 0

Since, coefficients of x are equal, we can directly subtract the equation.

Let's take another example. This time, we will solve linear equations in two variables.

Solving x + 2y = 3 and 2x + 4y - 6 = 0

1. Arranging the equations in the form ax + by = c
   
   \(x + 2y = 3 ⇒ equation \ 1 \\ 2x + 4y = 6 ⇒ equation \ 2\)
    
2. Multiplying equation 1 by 2:
   
   \((x + 2y) × 2 = 3 × 2 \\ 2x + 4y = 6\)
    
3. Subtract equation 1 from equation 2
   
   \((2x + 4y) - (2x + 4y) = 6 - 6 \\ 0 = 0\)

When two equations of lines are parallel, there are no possible solutions. Solving such equations gives two unequal numbers on both sides of the not equal sign. This eliminates both the variables. 

Let's understand it with an example of a linear equation in one variable. 
 

Equation 1 → 3(x - 2) + 4 = 0

Equation 2 → 3x - 5 = 0

Since, coefficients of x are equal, we can directly subtract the equation.

Now, let's take another example. This is a more advanced problem since, both the equations are of two variables. 

Suppose there are two lines of equation: x + 2y = 5 and 3x + 6y = 4.

• Step 1: Arranging equation:
  
  \(x + 2y = 5 ⇒ equation \ 1 \\ 3x + 6y = 4 ⇒ equation \ 2\)

• Step 2: Making the coefficients of x equal.  
  
  \(3 × (x + 2y = 5) \\ = 3x + 6y = 15 → equation \ 3\)

• Step 3: Subtracting equation 3 from equation 2
  
  \(⇒ 3x + 6y - (3x + 6y) = 15 - 4 \\ ⇒ 4 = 15\)

An equation with 3 variables is in the form Ax + By + Cz = D. We can use the elimination method to solve a system of 3 equations. We can learn it with an example. 

Solve the system of equations: x + y + z = 6,

\(2x - y + z = 3, \\ 3x + 2y - z = 4. \)

• Here, the equations are in the standard form: \(Ax + By + Cz = D\)

  
  \(x + y + z = 6        ⇒ 1\\ 2x - y + z = 3       ⇒ 2 \\ 3x + 2y - z = 4      ⇒3\)

• Eliminating z between equations 1 and 2:
  
  Subtract equation 1 from 2:
  
  \((2x - y + z) - (x + y + z) = 3 - 6 \\ x - 2y = -3 \\ x - 2y = -3    ⇒ 4\)

• Eliminating z in 1 and 3
  
  Adding 1 and 3 to eliminate z
  
  \((x + y + z) + (3x + 2y - z) = 6 + 4 \\ 4x + 3y = 10   ⇒5\)

• Now we have 2 equations with two variables
  
  \(x - 2y = -3           ⇒ 4 \\ 4x + 3y = 10       ⇒ 5\)

• Solving 4 and 5: 
  Multiplying equation 4 by 4:
  
  \(​​​​​​​(x - 2y = -3) × 4 \\ \quad 4x - 8y = -12 \)
  
  Subtract equation 5 to eliminate x
  
  \((4x - 8y) - (4x + 3y) = -12 - 10 \\ 0 - 11y = -22 \\ y = {-22 \over -11} = 2\)
  
   
• Substituting y in equation 4:
  
  \(x - 2(2) = -3 \\ x - 4 = -3 \\ x = -3 + 4 \\ x = 1\)
  
   
• Substituting x and y in equation 1 
  
  \(1 + 2 + z = 6  \\ 3 + z = 6\\ z = 6 - 3 \\ z = 3\)

Here, x = 1, y = 2, z = 3

The elimination method is used to solve two or more linear equations; it is not applicable to solve a single equation. A system of linear equations consists of two or more linear equations. The system of linear equations is enclosed in the symbol ‘{’
 

To solve a linear equation using the elimination method, we eliminate one variable to find its value.

• If the equations have the same coefficient of one variable, we simply add or subtract the equations.
   
• If there are no same coefficients, we multiply or divide the equations and then add or subtract them. 

To make calculations easy, here are a few tips and tricks for children. 

1. Always multiply or divide on both sides of the equations. 
2. You can only add or subtract coefficients of the same variables. For example, 2x + 3y can't be added. 
3. For multiplication, you can use multiplication tables. 
4. To check if the given equation of lines are parallel, use the condition: \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}\).
5. Verify the calculated values of variables by substituting it in the given equation.  

In real life, we use the elimination method in fields like economics, budgeting, physics, and so on. Here are some real-world applications of the elimination method:

1. To analyze the profit and cost, companies set up a system of equations, and by using the elimination method, we can determine the profit and cost.
   
    
2. In chemistry, we use elimination methods to determine the proportions of different compounds needed to achieve the desired mixture. 
   
    
3. In physics, to solve the problems related to force, acceleration, and motion, we use the elimination method. It is also applied in equations for electrical circuits.
   
    
4. In geometry, elimination method are used to find the common point between two lines. In computer graphics and robotics, such methods are used to create path intersection.
   
    
5. In machine learning and AI models, system of equations are used in algorithms such as in linear regression and neural training.