Question

Solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent. $$ \left\\{\begin{array}{l} 3 x-2 y+2 z=6 \\ 7 x-3 y+2 z=-1 \\ 2 x-3 y+4 z=0 \end{array}\right. $$

Short answer

Put your short answer here. Short answer must be a very short answer to the question based on the solution steps

Step-by-step solution

Write the system of equations as an augmented matrix

The given system of equations is:

Key concepts

augmented matrix

An augmented matrix is an efficient way to represent a system of linear equations. In an augmented matrix, each equation in the system is written in matrix form, and the constants on the right-hand side are included in the matrix as an extra column.
For example, consider the system:
\(\begin{array}{l} 3x - 2y + 2z = 6 \ 7x - 3y + 2z = -1 \ 2x - 3y + 4z = 0 \ \right. \)
The augmented matrix for this system is:
\[ \begin{pmatrix} 3 & -2 & 2 & | & 6 \ 7 & -3 & 2 & | & -1 \ 2 & -3 & 4 & | & 0 \ \end{pmatrix} \]
In this matrix, each row represents one equation, and each column (except the last one) represents the coefficients of one variable. The last column represents the constants from the right side of the equations. This matrix format makes it easier to perform row operations to solve the system.

system of equations

A system of equations is a collection of two or more equations with the same set of variables. The goal in solving a system is to find values for the variables that satisfy all the equations simultaneously. In our example, we're trying to find values for x, y, and z that satisfy:
\(\begin{array}{l} 3x - 2y + 2z = 6 \ 7x - 3y + 2z = -1 \ 2x - 3y + 4z = 0 \ \right. \)
Systems of equations can be solved using various methods, such as substitution, elimination, or matrix row operations. Matrix row operations are particularly useful for larger systems, as they allow us to systematically eliminate variables and isolate solutions.

inconsistent system

An inconsistent system is one that has no solution. This occurs when the equations represent parallel planes that never intersect or if the system requires contradictory conditions for the variables.
In our example, if during the process of row operations, we encounter a situation where we have a row like:

\[ \begin{pmatrix} 0 & 0 & 0 & | & c \ \end{pmatrix} \]
where c is a non-zero constant, this indicates an inconsistency. This row represents an equation that states 0 = c, which is impossible. Hence, it tells us that the system has no solution.

matrix solution steps

To solve a system of equations using matrix row operations, follow these steps:
1. **Write the system as an augmented matrix:** This represents the system in a structured form.
2. **Use row operations to simplify the matrix:** The three main row operations are:
* Swap the positions of two rows.
* Multiply a row by a non-zero scalar.
* Add or subtract the multiple of one row to another row.
3. **Continue these operations until the matrix is in row-echelon form:** This form makes the system easier to solve.
Example for our system:
\[ \begin{pmatrix} 3 & -2 & 2 & | & 6 \ 7 & -3 & 2 & | & -1 \ 2 & -3 & 4 & | & 0 \ \end{pmatrix} \]

After some row operations, we might get:
\[ \begin{pmatrix} 1 & 0 & 0 & | & x \ 0 & 1 & 0 & | & y \ 0 & 0 & 1 & | & z \ \end{pmatrix} \]
where x, y, and z are the solutions for the variables.