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Linear Systems

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Chapter #1                                                        

Linear Systems

        A linear system is a mathematical model of a system based on the use of a linear operator. It is in which the outputs are components of a vector which is equal to the value of a linear operator applied to a vector whose components are the inputs. It is where all the interrelationships among the quantities involved are expressed by linear equations which may be algebraic, differential, or integral.

Equivalent equations- Equations that have the same graph or solution.

Equivalent systems- Systems of equations that have the same solution.

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Parallel and distinct









Infinitely many

        The graphs of two linear equations in two variables may intersect at one point, be parallel and distinct, or coincide.


Substitution Method

        With the substitution method, we solve one of the equations for one variable in terms of the other, and then substitute that into the other equation.

2y + x = 3


4y – 3x = 1


Solve for x in equation (1)

2y + x = 3

x = 3 – 2y


Substitute (3-2y) for x in equation (2)

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class="c2">one of the variables.


Substitute 1 for y in equation (1)


Therefore the solution is (1, 1).


Elimination Method

        In the ‘elimination’ method for solving simultaneous equations, two equations are simplified by adding them or subtracting them. This eliminates one of the variables so that the other variable can be found.

2x - 5y =   1

3x + 5y = 14

Add the equations eliminates the y terms and will create an equation an equation with one variable

2x - 5y                         =


3x + 5y                         =



(2x - 5y) + (3x + 5y)     =

1 + 14

5x   =   15

x = 3

Substitute 3 for x in equation (1)

2x - 5y= 1

2(3) – 5y= 1

6 – 5y= 1

6 – 1= 5y

5= 5y

y= 1

Therefore the solution is (3, 1).

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