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Solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent. $$ \left\\{\begin{array}{l} 3 x-y=7 \\ 9 x-3 y=21 \end{array}\right. $$

Short Answer

Expert verified
The system has infinite solutions: \(x = t\) and \(y = 3t - 7\) for any real number \(t\).

Step by step solution

01

- Write the augmented matrix

First, convert the system of equations into an augmented matrix. For the equations: \[3x - y = 7\] and \[9x - 3y = 21\], the augmented matrix is: \[\begin{pmatrix} 3 & -1 & | & 7 \ 9 & -3 & | & 21 \end{pmatrix}\].
02

- Perform row operations to get the upper triangular form

Next, aim to form an upper triangular matrix using row operations. Start by making the element in the second row, first column (9) into zero by: \[R2 \rightarrow R2 - 3R1\]. This gives: \[\begin{pmatrix} 3 & -1 & | & 7 \ 0 & 0 & | & 0 \end{pmatrix}\].
03

- Interpret the resulting matrix

The resulting matrix \[\begin{pmatrix} 3 & -1 & | & 7 \ 0 & 0 & | & 0 \end{pmatrix}\] shows that the second equation simplifies to \[0 = 0\], indicating a dependent system (infinite solutions) instead of a contradiction.
04

- Expressing solution as parametric equations

Since there are infinite solutions, let \(x = t\) (where \(t\) is any real number). Substitute \(x = t\) into the first original equation to find \(y\): \[3t - y = 7\] gives \[y = 3t - 7\]. Thus, the solution is parameterized as: \(x = t\), \(y = 3t - 7\) for any real number \(t\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Augmented Matrix
An augmented matrix is a powerful tool used to solve systems of linear equations. It combines the coefficients of the variables and the constants from the equations into a single matrix. For the given system:
\[ \begin{array}{l} 3x-y=7 \ 9x-3y=21 \ \ \ \ \rightarrow \rightarrow \rightarrow \rightarrow \ \ \rightarrow \ \rightarrow \end{array} \]
we can represent it as:
\[ \begin{pmatrix} 3 & -1 & | & 7 \ 9 & -3 & | & 21 \end{pmatrix} \]
In this matrix, each row corresponds to an equation, each column corresponds to a coefficient of a variable, and the last column represents the constants on the other side of the equals sign. This representation makes it easier to apply row operations and find solutions.
Row Operations
Row operations are essential steps to manipulate matrices and find solutions to systems of equations. They include:
  • Swapping two rows
  • Multiplying a row by a nonzero scalar
  • Adding or subtracting a multiple of one row from another

In the original exercise, we used row operations to convert our augmented matrix into a simpler form:
1. Start with the initial matrix: \[ \begin{pmatrix} 3 & -1 & | & 7 \ 9 & -3 & | & 21 \ \end{pmatrix} \]
2. Make the element in the second row, first column zero by: \[ R2 \rightarrow R2 - 3 R1 \] which results in: \[ \begin{pmatrix} 3 & -1 & | & 7 \ 0 & 0 & | & 0 \end{pmatrix} \]
This step reduces the system to a simpler form, making it easier to interpret and solve.
Parametric Equations
Parametric equations provide a way to describe infinite solutions using one or more parameters. In our problem, we realized the system is dependent, indicating infinite solutions.

We let \( x = t \) where \( t \) is any real number. This allows us to express \( y \) in terms of \( t \):

From the equation \( 3t - y = 7 \), we solve for \( y \):
\[ y = 3 t - 7 \]
Therefore, the solutions to the system can be written as parametric equations:
  • \( x = t \)
  • \( y = 3 t - 7 \)

These equations represent an infinite set of solutions, where \( t \) can be any real number.
Dependent System
A dependent system of equations occurs when the equations describe the same line, resulting in infinite solutions. In our example, the second equation is a multiple of the first:

Transforming the system into an augmented matrix and performing row operations:
\[ \begin{pmatrix} 3 & -1 & | & 7 \ 9 & -3 & | & 21 \end{pmatrix} \rightarrow \ \begin{pmatrix} 3 & -1 & | & 7 \ 0 & 0 & | & 0 \end{pmatrix} \]
The second row indicates \( 0 = 0 \), showing that any solution for the first equation fits the second. This dependency means the system has infinitely many solutions, forming a dependent system.

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