System of Equations and Some Methods to Solve It

A set or collection of equations that are dealt with together are often regarded as a system of equations. These equations can be solved in two ways, either algebraically or graphically. A point where two points interact is the solution of the system of equations. In mathematics, a set of simultaneous equations is also known as the system of equations. System of equations has its origin from Europe with the introduction in 1637 by Rene Descartes. In this article, we will try to understand some concepts of the system of equations such as, methods to solve the system of equations and see some related examples.

Methods To Solve System of Equation

There are various ways through which you can solve the system of equations. Some of them are mentioned below:

Substitution Method: The method of substitution is one of the most convenient ways by which you can solve a linear equation. It is the process of solving one variable and then putting the value of it in the second equation in order to obtain the value of the second variable and solve the given system of equations. You can then check the solution in both the equations and verify your answer.

Elimination Method: The elimination method is also known as the addition method and is one of the ways which is often used in order to solve a linear equation. In this method, the two terms with the same variable are added to the opposite coefficient with a sum as zero. Then, the process of cross multiplication is used to eliminate one of the variables in the equation. Let us see the steps to use in the elimination method.

  • We have to arrange the variables to the left and the constants on the right.
  • Then, write the equation according to their respective variables.
  • Find if the variable in the first equation has the opposite coefficient in the second equation. If it exists, then add them and eliminate the variable.
  • If there are no opposite coefficients, then cross multiply a number to the equations to obtain one to add them and eliminate one variable.
  • Solve the equation of the other variable and find its value.
  • Substitute the values of the variable obtained in one of the equations and get the value of another variable and then check the solution.

Some Related Examples

Example 1:

Solve the system of equations 5x + 3y = 7, 3x – 5y = -23 by using the substitution method.

Given two equations are;

5x + 3y = 7

3x – 5y = -23

In the first equation, substitute 3y from both the sides,

5x + 3y – 3y = 7 – 3y

Divide both sides by 5

5x/5 = 7/5 – 3y/5, on simplification, x = 7- 3y / 5

Substitute x = 7 – 3y /5,

Substitute y=4, x= 7 – 3.4/5 = -5/5

= -1.

The solution of the given system of equation are as follows, x = -1 , y = 4

Example 2:

Solve the pair of equations 8x+5y = 23; 3x+2y = 9 by elimination method.

Given equations are 8x+5y = 23 (Equation 1) and

3x+2y = 9 (Equation 2)

Multiply Equation 1 by 3

24x+15y = 69 (Equation 3)

Multiply (Equation 2) by 8

24x+16y = 72 (Equation 4)

Subtract Equation 4 from Equation 3

24x+15y = 69

-24x-16y = -72

⇒-y = -3

⇒y = 3

Substituting the value of y in equation 2

3x+2×3 = 9

3x+6 = 9

3x = 9-6

3x = 3

⇒x = 3/3 = 1

Hence x = 1 and y = 3.

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