# How Should Solutions to a System of Linear Equations Be Expressed for Clarity?

• Kiefer
In summary, using a system of linear equations, I found three solutions: (x1)=9, (x2)=4, (x3)=0, (x4)=0, (x1)=4, (x2)=1, (x3)=1, (x4)=0, and (x1)=-1, (x2)=-2, (x3)=2, (x4)=0. These solutions represent points on a line that goes through the point <9, 4, 0, 0>. To represent the line in a parameter form, it is recommended to finish the row reduction to get the matrix in reduced row-echelon form and express it into its nullspace
Kiefer
Find all solutions to the following system of linear equations:
(x1) – 2(x2) – (x3)+(x4)=1
2(x1) – 3(x2) + (x3) – (x4)=6
3(x1) – 3(x2) + 6(x3))=15
(x1) + 5(x3)+(x4)=9

Using a system of linear equations, I found:
1 -2 -1 0 1
0 1 3 -3 4
0 0 0 6 0
0 0 0 0 0

so three solutions are:
(x1)=9, (x2)=4, (x3)=0, (x4)=0
(x1)=4, (x2)=1, (x3)=1, (x4)=0
(x1)=-1, (x2)=-2, (x3)=2, (x4)=0

How do I write my final solution (ie:what form)?

Kiefer said:
Find all solutions to the following system of linear equations:
(x1) – 2(x2) – (x3)+(x4)=1
2(x1) – 3(x2) + (x3) – (x4)=6
3(x1) – 3(x2) + 6(x3))=15
(x1) + 5(x3)+(x4)=9

Using a system of linear equations, I found:
1 -2 -1 0 1
0 1 3 -3 4
0 0 0 6 0
0 0 0 0 0

so three solutions are:
(x1)=9, (x2)=4, (x3)=0, (x4)=0
(x1)=4, (x2)=1, (x3)=1, (x4)=0
(x1)=-1, (x2)=-2, (x3)=2, (x4)=0

How do I write my final solution (ie:what form)?

Every solution is a point on a line that goes through <9, 4, 0, 0>. Your book should have some examples of representing lines with a parameter.

I would advise finishing your row reduction to get the matrix in reduced row-echelon form.

Express it into its nullspace (all the special solutions) and particular solution, if you've done that in linear algebra? The sum of the nullspace and particular solution gives the complete solution.

## What are system of linear equations?

A system of linear equations is a set of two or more linear equations with multiple variables. These equations can be solved simultaneously to find the values of the variables that satisfy all equations in the system.

## How do you solve a system of linear equations?

There are several methods to solve a system of linear equations, including substitution, elimination, and graphing. These methods involve manipulating the equations to isolate one variable and then substituting its value into the other equations to find the values of the remaining variables.

## What is the importance of system of linear equations?

System of linear equations is important because it allows us to model and solve real-world problems using mathematical equations. These problems can range from simple budgeting and financial planning to more complex engineering and scientific calculations.

## What is the difference between consistent and inconsistent systems of linear equations?

A consistent system of linear equations has at least one solution, meaning the lines representing the equations intersect at a single point. An inconsistent system has no solutions, meaning the lines are parallel and do not intersect.

## Can a system of linear equations have infinitely many solutions?

Yes, a system of linear equations can have infinitely many solutions. This occurs when the equations represent the same line or when they are dependent on each other. In this case, any point on the line satisfies all equations in the system.

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