A Guide On Algebra Elimination Method For Beginners

Have you heard the term “Linear Equations”? Of course, you had. In mathematics, it is the collection of two or more linear equations with two or more unknown variables. Solving the linear equation means you have to find the answers to the unknown variables. Well, there are several methods to solve a linear equation. For example, the graphical method, the substitution method, the elimination method, the cross-multiplication method, etc. Today we will learn how to solve linear equations with the help of an algebra elimination method.

However, out of all the methods for solving linear equations, the elimination method is the most popular. This method is simple and clear to use. Because it removes one of the terms that simplify our calculations.

Therefore, if you want to learn this algebra elimination method and algebra homework help, keep reading this blog. You will get essential information regarding the same. So, let’s get started.

Elimination Method

The elimination method means removing one of the terms containing any of the variables from the equation to make the calculations easier. You can do this method by multiplying or dividing a number by the equation(s). Until all of the variable terms’ coefficients are the same.

Note: Another name for the elimination method is the addition method. Because, to delete or remove that term from the result, we add or subtract both equations.

 

Remember To get the equation into one variable, either add or subtract the equations. We can add the equation to delete the variable if the coefficients of one of the variables are the same and the sign of the coefficients is the opposite. Similarly, we can subtract the equation to get the equation in one variable. If one of the variables’ coefficients is the same as the other’s, and the sign of the coefficients is the same. Moreover, if we do not have an equation to directly add or subtract the equations to remove the variable. We can start by multiplying one or both equations by a constant value on both sides of the equation. Then, we reduce the variable by simply adding or subtracting equations.

Now, let’s discuss the steps to solve linear equations with the elimination method.

Algebra Elimination Method: Steps To Solve Linear Equation

When solving linear equations with two or three variables, the elimination method comes into play. This method helps in solving three equations. But, at one time, you can use it to solve two equations. The following are the steps to solve a system of equations using the elimination method.

Step 1: To find a common coefficient of any one of the variables in both equations. You have to multiply or divide both linear equations with a non-zero value.

Step 2: Add and subtract both equations in such a way that similar terms are removed.

Step 3: Simplify the result to produce a final solution for the absent variable (or, y) in the form of y=c, where c is any number.

Step 4: Finally, use this value to find the value of the other given variable in another given equation.

However, this is the algebra elimination method to solve the two linear equations.

Let’s take an example and use the above steps to solve it. It will help you understand the method better. So, let’s take two equations. For example,

x + y = 8 and 2x – 3y = 4

Let,

x + y = 8 …(1)

2x – 3y = 4 …(2)

More Steps

Step 1: Multiply equation (1) by 2 and equation (2) by 1 to equal the coefficients of x. We get,

(x + y = 8) X 2 …(1)

(2x – 3y = 4) X 1 …(2)

So, now we have have two equations are,

2x + 2y = 16 …(1)

2x – 3y = 4 …(2)

Step 2: Subtract the equations.

 2x + 2y = 16

 2x – 3y = 4

• +      –

5y = 12

Thus, we have y = 12/5.

Step 3: Put the value of y in equation 1.

x + y = 8

X = 8 – y

x = 8 – 12/5

x = 28/5

Therefore, now we have x = 28/5 and y = 12/5.

But what if we get the values of both variables equal while multiplying a non-zero constant? What if, in adding or subtracting, both terms were removed? When solving equations with parallel and coinciding lines, we get similar cases.

Equations involving two intersecting lines have only two consistent solutions. But equations involving two parallel lines have no solutions. Because these lines never intersect. So, their equations have an endless number of solutions. Confused? Let’s discuss both situations in detail.

Algebra Elimination Method: No Solution

There are no solutions to equations with two parallel lines. So, when we use the elimination method to solve these equations, we get the result as two unequal values. For example, 0 is not equal to 8, and 0 is not equal to -2. 

However, in this case, we can not remove one variable. Both variables will be removed from the equation.

For example, we have equations,

2x – y = 4 …(1)

4x – 2y = 7 …(2)

So, we multiply equation (1) by 2 and equation (2) by 1 to make the x coefficients in both equations equal.

(2x – y = 4) X 2

(4x – 2y = 7) X 1

Therefore, we get, 

4x – 2y = 8 …(3)

4x – 2y = 7 …(4)

Now, we will subtract both equations.

4x – 2y = 8

 4x – 2y = 7

•  +       –

We get, 0 = 1. 

Because the results are different, there is no alternative way to solve these problems. Therefore, when there is no solution, we get this result using the elimination method.

Now, let’s discuss the other situation.

Algebra Elimination Method: Infinitely Many Solutions

There are an infinite number of solutions for two equations with coinciding lines. So, if we solve a system of equations with coincident lines using the elimination method. We will get a consistent system with infinite solutions. If we use the elimination method in these situations, we get a result in the form of 0=0.

Let’s take an example, 

We have equations, x + y = 2 and 2x + 2y – 4 = 0.

So, when you multiply any non-zero constant with both equations. You will see that the x- and y-variable components are cancelled or eliminated every time.

Let’s have a look,

x + y = 2 …(1)

2x + 2y – 4 = 0 …(2)

Let’s multiply the 1st equation by 2 and the 2nd with 1.

(x + y = 2) X 2

(2x + 24 – 4 = 0) X 1

We get, 

2x + 2y = 4 …(3)

2x + 2y – 4 = 0 …(4)

Now, we will subtract both the equations (3) and (4).

2x + 2y = 4

2x + 2y – 4 = 0

We will get 0 = 0.

Therefore, when there are infinitely many solutions, we get a result in the form of 0=0 using the elimination method. However, before trying to solve the above linear equations, make sure they have:

• Intersecting Lines
• Parallel Lines
• Coinciding Lines

Important Points To Remember

• The algebra elimination method helps to solve the system of equations.
• This method is simple. Also, it reduces the calculation by reducing one variable.
• To delete the related variable, we make the coefficient of a variable the same.

Final Words

To sum up, in the above blog we have learned how to solve linear equations with an algebra elimination method. It is a simple method to solve linear equations. From the above-mentioned steps, you can solve more linear equations. Moreover, to master this method, the only tip is to practice. In case you are still confused about this method, you should hire a private tutor. They will help you understand the method more efficiently.