Elimination Method of Solving a Pair of Linear Equations

While solving an equation the most common method that is used is the elimination method. It is based on eliminating one variable to solve the equation. It is also popularly known as the addition method. 

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How to solve an equation through the elimination method?

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To solve an equation with the elimination method first multiply the constant with the help of a constant. 

Step by step guide

Here is a step-by-step guide on how to use the elimination method for solving an equation:

1. The first step includes multiplication of both the linear equations by a constant on a non-zero value, making the coefficients of either of the variables x or y (numerically equal)
2. Now adding or subtracting one equation from the other in such a way that the variables are easily eliminated. Once an equation is obtained with one variable the following steps can be followed easily. There are two possibilities if you do not get to this step:
3.  A true statement with no variable meaning that the original equations have infinite solutions.
4. A false statement with no variable meaning the original equations do not have any solution and are inconsistent.
5. Solving the equation two variables either x or y, now you would get the value of that specific value.
6. At last substitute the value in the previous equation, you can now get the value of another variable as well. This will help you solve elimination method problems.

For better understanding

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 = -4

y = -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

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Elimination method examples

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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?

Supposedly let's assume that you are to solve word problems without the equations. How will you solve the problem with the help of the elimination method?

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 x to 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 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.