⎧⎪⎨⎪⎩2x−4y+z=7−x+y−z=−4x−2y=3


We are given a system of three linear equations:

\[
\begin{cases}
2x - 4y + z = 7 \\
-x + y - z = -4 \\
x - 2y = 3
\end{cases}
\]

### Step 1: Solve the third equation for \( x \)

The third equation is:
\[
x - 2y = 3
\]
Solving for \( x \):
\[
x = 2y + 3
\]

### Step 2: Substitute \( x = 2y + 3 \) into the first and second equations

#### Substituting into the first equation:
\[
2(2y + 3) - 4y + z = 7
\]
Simplifying:
\[
4y + 6 - 4y + z = 7 \quad \Rightarrow \quad z = 1
\]

#### Substituting into the second equation:
\[
-(2y + 3) + y - z = -4
\]
Substitute \( z = 1 \):
\[
-(2y + 3) + y - 1 = -4
\]
Simplifying:
\[
-2y - 3 + y - 1 = -4 \quad \Rightarrow \quad -y - 4 = -4 \quad \Rightarrow \quad y = 0
\]

### Step 3: Substitute \( y = 0 \) into \( x = 2y + 3 \)

Substitute \( y = 0 \) into \( x = 2y + 3 \):
\[
x = 2(0) + 3 = 3
\]

### Step 4: Conclusion

The solution to the system is:
\[
x = 3, \quad y = 0, \quad z = 1
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you solve a 3x3 system of equations using matrix methods?
2. Can you solve a system of equations graphically?
3. How do substitution and elimination methods compare for solving systems of equations?
4. What are some applications of systems of equations in real life?
5. How can systems of equations be solved using Gaussian elimination?

**Tip:** Always check your solution by substituting the values of variables back into the original equations!

Learn how to solve a system of three linear equations using the substitution method. This problem involves algebraic concepts like solving for variables, substitution, and systems of linear equations. Step-by-step instructions are provided, making it suitable for students in Grades 9-12.

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