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Systems of Equations

A system of equations is a set or a group of equations. System of equations is a collection of two or more equations with the same set of variables. Equations in a system can be linear or non-linear. But usually, system of equations refers to a set of linear equations with same variables. By linear equation, we mean an equation of degree "one".
An example of system of linear equation is:
x - y + z = 2
2x + y - z = 1
x - 3y + 2z = -3
Properties of System of Equations
The important properties of a system of linear equations are:

  • Consistency - A consistent system of equations possesses a set of solution and a system of equation which has no solution is called an inconsistent system of equation.
  • Equivalence - Two systems of equations are said to be equivalent to each other, if they have same set of solutions.
  • Independence - Each equation in a system is independent, i.e. it can not be derived from any other equation in the system.
Solving System of Equations
The values of all variables contained in a system of equations is known as its solution. There are various methods of solving a system of linear equations:
  • Elimination Method: This method is based on eliminating one variable and finding the value of other.
  • Substitution Method: According to this method, an algebraic expression for one variable is determined from an equation and substituted into another.
  • Graphical Method: In this method, we draw graphs for all equations and find the common intersection point as a solution.
Let us consider an example:
Example: Solve the following system of equations-
2x + 3y = 4
x - 2y = 1
Solution: Let us use substitution method
2x + 3y = 4 ................(1)
x - 2y = 1 ...................(2)
Find the value of x from equation (2)
x = 1 + 2y ..................(3)
Substitute this value in equation (1)
2(1 + 2y) + 3y = 4
2 + 7y = 4
7y = 2

y = $\frac{2}{7}$

Plug the value of x in equation (3)

x = 1 + 2($\frac{2}{7}$)

x = 1 + $\frac{4}{7}$

x = $\frac{11}{7}$

Therefore, the solution is:

x = $\frac{11}{7}$

y = $\frac{2}{7}$

Related Calculators
Linear System of Equations Solver
 

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