Write out the augmented matrix for the following linear...

Write out the augmented matrix for the following linear systems. Then solve the system by first applying elementary row operations of type \#1 to place the augmented matrix in upper triangular form, followed by Back Substitution. (a) $$ \begin{array}{r} x_1+7 x_2=4 \\ -2 x_1-9 x_2=2 . \end{array} $$ (b) $$ \begin{aligned} 3 z-5 w & =-1 \\ 2 z+w & =8 . \end{aligned} $$ (c) $$ x-2 y+z=0, $$ $$ 2 y-8 z=8 \text {, } $$ $$ -4 x+5 y+9 z=-9 \text {. } $$ $$ p+4 q-2 r=1, $$ $$ x_1-2 x_3=-1 \text {, } $$ $$ -x+3 y-z+w=-2, $$ $$ x_2-x_4=2 \text {, } $$ (d) $\quad-2 p-3 r=-7$, $$ 3 p-2 q+2 r=-1 \text {. } $$ (e) $$ \begin{aligned} x-y+3 z-w & =0, \\ y-z+4 w & =7, \\ 4 x-y+z & =5 . \end{aligned} $$

Write out the augmented matrix for the following linear systems. Then solve the system by first applying elementary row operations of type \#1 to place the augmented matrix in upper triangular form, followed by Back Substitution.
(a)
$$
\begin{array}{r}
x_1+7 x_2=4 \\
-2 x_1-9 x_2=2 .
\end{array}
$$
(b)
$$
\begin{aligned}
3 z-5 w & =-1 \\
2 z+w & =8 .
\end{aligned}
$$
(c)
$$
x-2 y+z=0,
$$
$$
2 y-8 z=8 \text {, }
$$
$$
-4 x+5 y+9 z=-9 \text {. }
$$
$$
p+4 q-2 r=1,
$$
$$
x_1-2 x_3=-1 \text {, }
$$
$$
-x+3 y-z+w=-2,
$$
$$
x_2-x_4=2 \text {, }
$$
(d) $\quad-2 p-3 r=-7$,
$$
3 p-2 q+2 r=-1 \text {. }
$$
(e)
$$
\begin{aligned}
x-y+3 z-w & =0, \\
y-z+4 w & =7, \\
4 x-y+z & =5 .
\end{aligned}
$$

Key Concepts

Back Substitution

Back substitution is a method used after transforming the augmented matrix into an upper triangular form, where one starts solving from the bottom-most equation and works upwards. With the system reduced to a form where each equation contains only one more unknown than the one below, the unknowns can be determined sequentially in a clear and efficient manner.

Gaussian Elimination

Gaussian elimination is a systematic method for solving systems of linear equations that involves using elementary row operations to transform the augmented matrix into an upper triangular or row echelon form. This transformation simplifies the structure of the system, making the successive elimination of variables straightforward.

Upper Triangular Form

An upper triangular form, or row echelon form, of an augmented matrix is one in which all entries below the main diagonal are zero. This form allows for a clear and systematic approach to solving the system by reducing it to a simpler, step-by-step problem where each equation involves one fewer variable than the previous.

Elementary Row Operations

Elementary row operations are the fundamental manipulations allowed on the rows of a matrix that do not alter the solution set of the corresponding system of linear equations. These operations include swapping rows, multiplying a row by a nonzero scalar, and adding a multiple of one row to another. They are crucial for transforming the matrix into simpler forms.

Augmented Matrix

An augmented matrix is a compact representation of a system of linear equations that combines the coefficients of the variables and the constant terms into a single matrix. This format is essential because it consolidates all relevant numerical information needed for systematic solution methods, making it easier to apply algorithmic techniques such as Gaussian elimination.

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