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solve it in normal way not augmented matrix

To solve the system of equations without using an augmented matrix, let's rewrite and solve it step-by-step. The given system is:

\[
\begin{cases}
x_2 + 4x_3 = -5 \\
x_1 + 3x_2 + 5x_3 = -2 \\
3x_1 + 7x_2 + 7x_3 = 6
\end{cases}
\]

### Step 1: Solve for \( x_2 \) in terms of \( x_3 \) from the first equation

From the first equation:
\[
x_2 + 4x_3 = -5
\]
\[
x_2 = -5 - 4x_3
\]

### Step 2: Substitute \( x_2 = -5 - 4x_3 \) into the second and third equations

Substitute into the second equation:
\[
x_1 + 3(-5 - 4x_3) + 5x_3 = -2
\]
Simplify:
\[
x_1 - 15 - 12x_3 + 5x_3 = -2
\]
\[
x_1 - 15 - 7x_3 = -2
\]
\[
x_1 = 13 + 7x_3
\]

Now, substitute \( x_2 = -5 - 4x_3 \) into the third equation:
\[
3x_1 + 7(-5 - 4x_3) + 7x_3 = 6
\]
Simplify:
\[
3x_1 - 35 - 28x_3 + 7x_3 = 6
\]
\[
3x_1 - 35 - 21x_3 = 6
\]
\[
3x_1 = 41 + 21x_3
\]
Substitute \( x_1 = 13 + 7x_3

Solve the following system of equations:

\[
\begin{cases}
x_2 + 4x_3 = -5 \\
x_1 + 3x_2 + 5x_3 = -2 \\
3x_1 + 7x_2 + 7x_3 = 6
\end{cases}
\]

Solve it in the normal way, not using an augmented matrix.

Learn how to solve a system of linear equations without using an augmented matrix. This problem explores substitution methods to solve for variables in equations such as x_2 + 4x_3 = -5, x_1 + 3x_2 + 5x_3 = -2, and 3x_1 + 7x_2 + 7x_3 = 6. Suitable for students studying college-level linear algebra or advanced high school algebra.

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