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Systems of Equations

A "system" of equations is a set or group of equations. Linear equations are easy than non-linear equations, and the simplest linear system is one with two equations and two variables.

System of equations is a collection of two or additional equations with a same set of unknowns. In solving a system of equations, we need to find values for every of the unknowns that will declare every equation in the system. The system of equation can be linear or non-linear. The problem can be spoken in sequence of actions form or the problem can be expressed in algebraic form.

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Methods of Systems of Equations

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Elimination:

Elimination technique is considered as one of the algebraic method for solving systems. In elimination method an operation on 1 equation is performed so that cancelling one variable and finding the other variable.

Substitution:

In substitution method the algebraic expression of one of the variable is substituted in another equation at the place of the respective variable and then the variable is solved. Again by substituting in any of the equation the value of the second variable is also found.

Linear equation:

An algebraic expression which relate two variables and whose graph is a line.

Matrix:

A rectangular array of number written in brackets and used to find solutions for complex systems of equations.

Graphical Method:
Graphical method is also one of the methods to find the solution of the system of equations by drawing graph of each equation on same plane.

Below are the examples on solving systems of equations

Example 1:

Given:

x + 27 = 71

Solution:

Step 1: x + 27 - 27 = 71 - 21 (Subtract 27 on both the sides)

Step 2: x = 50 (So the answer is 50)

Example 2:

Given

a + 15 = 75.

Solution :

Step 1: We need to find the value of a.

Step 2: Subtract 15 on both the sides.

Step 3: So the value of a is 60.

Example 3:

y = 2x + 1, 2y = 3x - 2

Solution:

Step 1: Substitute one equation to another. So 2(2x + 1) = 3x - 2.

Step2: Now we have single variable equation, we need to solve that variable equation.

x = 2 = -2

So the value of x = -4

Step 3: Now we solved that variable .so now we need to place in that variable back into either equation to obtain the value of y at the solution.

We know that x = -4.Now we need to find the y value y = 2(-4) + 1 = -7.

So x = -4, y = -7

Example 4:

Solve the following equations by using the Elimination method:

2x + 2y = 4

4x - 2y = 8

Solution:

Step 1:

2x + 2y = 4.

4x - 2y = 8.

Step 2:

Subtracting equation 2 from 1 ,we get.

6x = 12

Step 3: Divide using 6 on both the sides so x = 2.

Step 4: Now plug x values in equation 1 .

4x - 2y = 8.

4(2) - 2y = 8.

8 - 2y = 8.

Y= 0.

Step 5: So, the solution is (2, 0)

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