Solving Systems of Equations: Finding General Solutions with Augmented Matrices

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In summary, a system of equations is a set of two or more equations with multiple variables that can be solved simultaneously. To solve using augmented matrices, the equations must be written in matrix form and row operations must be performed. The general solution of a system of equations is the set of all possible solutions expressed in terms of parameters or variables. A system of equations has no solution if the augmented matrix results in contradictory equations. However, a system of equations can have more than one solution if the equations are not contradictory, resulting in an infinite number of solutions.
  • #1
SMA_01
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Homework Statement



The question says to find the general solutions of the systems, given the augmented matrices:

1 3 4 l 7
3 9 7 l 6



Homework Equations





The Attempt at a Solution



1 3 4 l 7
3 9 7 l 6 --> (-3)1st+2nd and simplified

and got:

1 3 4 l 7
0 0 -5 l -15

(-3)1st+2nd and simplified then got:

1 3 4 l 7
0 0 1 l 3

and finally, (-4)2nd+1st:

1 3 0 l -5
0 0 1 l 3


Does this look right? I'm just wondering if I got the correct answer. Thanks
 
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  • #2
You correctly ranked the matrix, but where is your solution?
It should be something like: (a,b,c) where a,b,c could be numbers or parameters.
 

FAQ: Solving Systems of Equations: Finding General Solutions with Augmented Matrices

What is a system of equations?

A system of equations is a set of two or more equations that contain multiple variables. The solutions to a system of equations are the values of the variables that satisfy all of the equations simultaneously.

How do I solve a system of equations using augmented matrices?

To solve a system of equations using augmented matrices, you must first write the system of equations in matrix form. Then, perform row operations on the augmented matrix until it is in reduced row-echelon form. The solutions to the system of equations can be found by looking at the values of the variables in the reduced row-echelon form.

What is the general solution of a system of equations?

The general solution of a system of equations is the set of all possible solutions that satisfy the equations. It can be expressed in terms of parameters or variables, depending on the number of equations and variables in the system.

How do I know if a system of equations has no solution?

A system of equations has no solution if, after performing row operations on the augmented matrix, you end up with a row of zeros on the left side of the vertical line in the augmented matrix and a non-zero number on the right side. This means that the equations are contradictory and cannot be satisfied simultaneously.

Can a system of equations have more than one solution?

Yes, a system of equations can have more than one solution. This occurs when the equations are not contradictory but can be satisfied by different values of the variables. These types of systems are referred to as consistent and have an infinite number of solutions.

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