Solving Systems of Equations


The other name for the system of equations is a set of simultaneous equations or an equation system, which is a finite collection of equations for which we sought common solutions in mathematics. A system of equations could be categorized in the same way as a single equation can. In our daily lives, systems of equations are used in modeling issues in which the unknown values may be represented in the form of variables.

solving systems of equations

In this article, we are going to know about what equations are & methods of solving a system of equations.

Definition of Solving Systems of Equations 

As previously established, a system of equations is defined as a collection of equations & all these systems of equations need a common solution for all the variables which are there in the equation. The set of equations below is an example of a system of equations.

3x – y = 24

2x-y = 12

Importance of Solving System of Equations

Solving an equation system entail determining the values of variables employed in the set or system of equations. We calculate the value of an unknown variable for example –  x & y while keeping the equations on both sides balanced. Do you know the basic goal behind solving equations? The main reason why we solve an equation system is to determine or to obtain the value of the variable (value of variables like x, y, z, etc.) that makes all of the supplied equations true. Any given system of equations might have a variety of solutions.

  • Solution which is unique.
  • There is no solution for the set of equations.
  • There are infinitely many solutions.

How to Solve System of Equations?

Different approaches can be used to solve any system of equations. We require at least two equations to solve a system of equations in two variables. Similarly, we will need at least three equations to solve a system of equations in three variables.

Let’s try to solve the given equation using the Substitution Method.

Example: Solve the two equations given below. 

            2x + 4y =16

            4x + 2y = 20

Solution:

Let’s try to find the values of x, y using the substitution method,

In the first equation, since the number 2 is common -> 2( x+ 2y =8) – – – (1)

In the second equation, since the number 2 is common here -> 2(2x+ y = 10) – – – (2)

Now,  x = 8-2y we can substitute the value of x in Equation (2)

2(8-2y) +y = 10

16 – 4y + y = 10 , 16 -3y = 10

Therefore, -3y = 10-16 ; -3y = -6 which equals y = 6/3 = 2

Now, we have the value of y as 2, let’s find the value of x by substituting the value of the variable y in any of the equations.

Putting the value of y in Equation (1), x+ 2*2 =8 , x+4 =8 

Therefore, the value of x = 8-4 = 4

Now, we have the values of x = 4 and y = 2.

There are many more methods by which you can find the value of the variables in any given set of equations.

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