Elimination Method

You know that equations of two parallel lines do not have a solution. Therefore, if we solve any such equation with the elimination method we will get two unequal numbers on both sides of the unequal sign. For example, 0≠8, -2≠0, etc. In such cases, we do not eliminate one variable but we eliminate both variables. An example is given below to illustrate this further.

Let us take two equations-

2x-y=4 __ (1) and 4x-2y=7 __ (2) and solve them by using the elimination method.

To make the x coefficients equal in both the equations, we multiply equation (1) by 2 and equation (2) by 1. We will get 4x-2y=8 __ (3) and 4x-2y=7 __ (4). 

Now, if we want to subtract equation 4 from equation 3, we get 0=1 because both the variables are getting eliminated. We cannot solve these equations by any other method because the solutions are inconsistent. Thus, if there is no solution in the elimination method, we get the result in this form.

The elimination method is one of the most commonly used methods when it comes to solving an equation. You can eliminate one variable so you can solve the equation with ease. It is also called the addition method. Here is how you can solve an equation with the help of the elimination method.In the elimination method, you first multiply the constant with the help of a constant. There are two ways in which you can solve a linear equation. One is by plotting a graph of the equations, and in the other one, you solve the equations through the elimination method.

Solving a Set of Linear Equations by the Elimination Method

For solving the simultaneous linear equations, you reduce the pair of the equations to one linear equation in one variable. You then solve it normally. Solving equations through the elimination method would require you to multiply the coefficients of the given equations and make the coefficients of one variable the same amongst both the equations. Lastly, just one variable would remain in the end which you can solve easily. Given below are the steps to solve by elimination method.

Steps for Solving Equations through the Elimination Method

Let’s look at how equations can be solved through the elimination method math step by step.

1. The first step is to multiply both the linear equations by a constant on a non-zero value. This would make the coefficients of either of the variables, x or y, numerically equal.

2. The next step is adding or subtracting one equation from the other in a way that one of the variables is easily eliminated. Once you get an equation with one variable, follow the next steps. If you do not get this, then there can be two possibilities:

• If you get a true statement with no variable, then it means that the original equations       have infinite solutions.

• If you get a false statement with no variable, then it means that the original equations do not have any solution and are inconsistent.

3. The next step is solving the equation with one variable, either x or y, and you would get the value of that specific value.

4.  substituting this value in the previous equation, you would get the value of the other variable as well. 

This will help you to solve the elimination method problems.

To Make it Simpler for You, Consider the Following Example

Solve this set of equations 2x + y = -4 and 5x – 3y = 1 using elimination method.

The equations given are: 2x + y = -4       …………… (i)

 5x – 3y = 1       …………… (ii)

 Multiplying equation (i) by 3, you get,

{2x + y = -4} …………… {× 3}

6x + 3y = -12       …………… (iii) 

Adding equations (ii) and (iii), you get, 5x - 3y = 1

6x + 3y = -12

____________

11x       = -11

____________

x = -11/11

Hence, x = -1

Substituting this value of x = -1 in equation (i), you get,

2 × (-1) + y = -4

-2 + y = -4y = -4 + 2

Hence, y = -2

Therefore, x = -1 and y = -2 is the solution of the set of equations 2x + y = -4 and 5x – 3y = 1

Elimination Method Examples

Take a look at the elimination method questions.

Example 1:

Solve the following equations using the addition method.

2x + y = 9

3x – y = 16 

Solution: 

If you add down, the y variables will cancel out.

2x + y = 9

3x – y = 16

__________

5x      = 25

__________

Substituting the value of x,

2(5) + y = 9

10 + y = 9

y = –1

Then the solution is x=5 and y=-1.

How to Solve Word Problems with the Elimination Method?

Suppose you are given a word problem without the equations. How do you solve the problem with the help of the elimination method? Well, the solution is simple. Take a look at the following example:

1. The charges of a park are \[$\] 10 for adults and \[$\] 5 for kids. How many adult tickets and kids tickets were sold if 548 tickets were sold for a total of \[$\] 3750?

First Step:

Consider the variable to x be the number of adult tickets and the variable y to be the number of kids tickets.

According to the problem, you can form two different equations,

x + y = 548 ---------(1)

And,

10x + 5y = 3750

Dividing both the sides by 5, you get,

2x + y = 750 --------(2)

Second Step:

The next step is to eliminate one of the variables for getting the value of the other variable.

In the equations (1) and (2), variable y is having the same coefficient. But it has the same sign in both the equations. 

For changing the sign of y in equation (1), multiply both sides of (1) by the negative sign. This will give you,

- (x + y) = - 548

- x - y = - 548 --------(3)

Third Step: 

Now, the step is eliminating the variable y in both the equations (2) and (3) as given below and finding the value of x. When you do so, you get,

2x + y = 750

-x -y = - 548

__________

x       = 202

__________

Fourth Step: 

Next, substitute the value 202 for x in equation (1) to get the value of the variable y. After doing so, you get,

202 + y = 548 Subtracting 202 from both sides, you get, 

y = 346

Therefore, the number of tickets sold for adults is 202 and the number of tickets sold for kids is 346.