To find out which type of matrix can be formed using the given system of linear equations, we need to set up the matrix including both the coefficients of the variables and the constants on the right-hand side of the equations.
Let’s write down the given system of equations:
1. [tex]\( 3x - 2y = -2 \)[/tex]
2. [tex]\( 7x + 3y = 26 \)[/tex]
3. [tex]\( -x - 11y = -46 \)[/tex]
### Step-by-Step Construction of the Matrix
1. Identify Coefficients and Constants:
- From the first equation [tex]\( 3x - 2y = -2 \)[/tex]:
Coefficients of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are [tex]\( 3 \)[/tex] and [tex]\( -2 \)[/tex], with a constant [tex]\( -2 \)[/tex].
- From the second equation [tex]\( 7x + 3y = 26 \)[/tex]:
Coefficients of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are [tex]\( 7 \)[/tex] and [tex]\( 3 \)[/tex], with a constant [tex]\( 26 \)[/tex].
- From the third equation [tex]\( -x - 11y = -46 \)[/tex]:
Coefficients of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are [tex]\( -1 \)[/tex] and [tex]\( -11 \)[/tex], with a constant [tex]\( -46 \)[/tex].
2. Form the Augmented Matrix:
An augmented matrix includes all the coefficients and the constants from the right-hand side of each equation.
[tex]\[
\begin{bmatrix}
3 & -2 & -2 \\
7 & 3 & 26 \\
-1 & -11 & -46
\end{bmatrix}
\][/tex]
3. Determine the Shape of the Matrix:
- Count the number of rows: There are 3 equations, so the matrix has 3 rows.
- Count the number of columns: Each row includes coefficients for [tex]\( x \)[/tex] and [tex]\( y \)[/tex], plus the constant term, making a total of 3 columns.
### Conclusion:
The matrix formed from this system of linear equations has 3 rows and 3 columns. This type of matrix is known as a square matrix because it has an equal number of rows and columns.
Therefore, the answer is:
- a square matrix
