use gauss reduction to solve: x+y+z=15,2x+y+4z=100,x+y+2z=60

We are given the system of linear equations:

\[
\begin{aligned}
1. \quad & x + y + z = 15 \\
2. \quad & 2x + y + 4z = 100 \\
3. \quad & x + y + 2z = 60
\end{aligned}
\]

### Step 1: Write the augmented matrix
\[
\begin{pmatrix}
1 & 1 & 1 & | & 15 \\
2 & 1 & 4 & | & 100 \\
1 & 1 & 2 & | & 60
\end{pmatrix}
\]

### Step 2: Perform Gaussian elimination

**Subtract Row 1 from Row 3** to eliminate the first element in Row 3:
\[
R_3 \rightarrow R_3 - R_1: \quad (1\ 1\ 2\ | \ 60) - (1\ 1\ 1\ | \ 15) = (0\ 0\ 1\ | \ 45)
\]
The matrix becomes:
\[
\begin{pmatrix}
1 & 1 & 1 & | & 15 \\
2 & 1 & 4 & | & 100 \\
0 & 0 & 1 & | & 45
\end{pmatrix}
\]

**Subtract 2 × Row 1 from Row 2** to eliminate the first element in Row 2:
\[
R_2 \rightarrow R_2 - 2R_1: \quad (2\ 1\ 4\ | \ 100) - 2(1\ 1\ 1\ | \ 15) = (0\ -1\ 2\ | \ 70)
\]
The matrix now looks like:
\[
\begin{pmatrix}
1 & 1 & 1 & | & 15 \\
0 & -1 & 2 & | & 70 \\
0 & 0 & 1 & | & 45
\end{pmatrix}
\]

### Step 3: Back substitution

We now solve the system by back substitution:
- From Row 3, we see \( z = 45 \).
- Substitute \( z = 45 \) into Row 2:
  \[
  -y + 2(45) = 70 \quad \Rightarrow \quad -y + 90 = 70 \quad \Rightarrow \quad -y = -20 \quad \Rightarrow \quad y = 20
  \]
- Substitute \( y = 20 \) and \( z = 45 \) into Row 1:
  \[
  x + 20 + 45 = 15 \quad \Rightarrow \quad x + 65 = 15 \quad \Rightarrow \quad x = -50
  \]

### Final Solution:
\[
x = -50, \quad y = 20, \quad z = 45
\]

Would you like more details on any step or have any questions?

Here are some related problems for practice:
1. Solve the system using Gaussian elimination: \( 2x + 3y = 13 \), \( 4x - y = 9 \).
2. Use Gauss-Jordan elimination for \( x + 2y + 3z = 6 \), \( 2x + y - z = 3 \), \( 3x - y + 2z = 7 \).
3. Solve the system \( 5x + 6y = 18 \), \( 3x + 2y = 7 \) using matrix inversion.
4. What is the solution to \( x + y = 10 \), \( x - y = 2 \)?
5. Apply Gaussian elimination to the system \( 3x + 4y = 25 \), \( 2x - 3y = 7 \).

**Tip:** When performing Gaussian elimination, always aim to simplify each row to help isolate variables step by step.

Learn how to solve a system of linear equations using Gaussian elimination. This detailed step-by-step solution covers key algebraic methods, including augmented matrices and back substitution, applied to the system x + y + z = 15, 2x + y + 4z = 100, and x + y + 2z = 60. Suitable for high school and early college students.

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