Linear algebra: solving questions that has 2 systems in it

In summary, the question is asking to solve two systems of equations simultaneously by using elimination on a 3x5 augmented matrix and performing two back substitutions.
  • #1
mroldboy
7
0

Homework Statement





solve the 2 systems

x1 + 2x2 - 2x3 =1 x1 + 2x2 - 2x3 = 9
2x1 + 5x2 +x3 = 9 2x1 + 5x2 +x3 = 9
x1 + 3x2 + 4x3 = 9 x1 + 3x2 + 4x3 = -2

the question gives this explanation

by doing elimination on a 3x5 augmented matrix and the performing two back substitutions.



Homework Equations



I don't think I understand what this question is asking, if someone could explain what it means with its explanation it would help a lot.

This is my first week taking linear algebra as well.


The Attempt at a Solution



I thought it meant to put the two systems together, but since the 2nd row of equations are the same there would only be 5, not 6 I would use so it would be 3x5.

1 2 -2 1
2 5 1 9
1 3 4 9
1 2 -2 9
1 3 4 -2

but then that would make 2 rows
0 0 0 x
and then wouldn't it be inconsistent. But the book has answers to the question so its not that. What is this question even asking me to do? Sorry if didn't put the matrix in the right form for on here.
 
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  • #2
The only difference between the two systems is the constants on the righthand side of the equations. Because the coefficient matrix is identical for both systems, you'd find you would apply exactly the same row operations to find each solution, so it makes sense just to reduce both systems at the same time using the matrix

[tex]
\left(\begin{array}{ccc|cc}
1 & 2 & -2 & 1 & 9 \\
2 & 5 & 1 & 9 & 9 \\
1 & 3 & 4 & 9 & -2
\end{array}\right)
[/tex]

The fourth column consists of the constants from the first system, and the fifth column consists of the constants from the second system.
 

1. What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of linear equations and their representations in vector spaces. It involves the study of systems of linear equations, matrices, vectors, and linear transformations.

2. What is a system of linear equations?

A system of linear equations is a set of equations that involve linear functions of the same set of variables. The goal is to find the values of the variables that satisfy all of the equations in the system simultaneously.

3. How do you solve a system of linear equations?

To solve a system of linear equations, you can use various methods such as Gaussian elimination, substitution, or the Cramer's rule. These methods involve manipulating the equations to eliminate variables and find the values that satisfy all of the equations.

4. What does it mean to have 2 systems in linear algebra?

In linear algebra, having 2 systems refers to having two sets of linear equations that need to be solved simultaneously. This means finding the values of the variables that satisfy both systems of equations at the same time.

5. How is linear algebra used in real life?

Linear algebra has many applications in real life, such as in engineering, physics, economics, and computer graphics. It is used to solve systems of equations that arise in these fields and to model and analyze real-world problems involving linear relationships.

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