To solve the system of equations:  
1. \(-4x + y = -29\)  
2. \(-x + y = -2\),  

we can use the **substitution** or **elimination method**. Here, we will use the **elimination method**.

---

### Step 1: Align equations for elimination
\[
-4x + y = -29 \tag{1}
\]
\[
-x + y = -2 \tag{2}
\]

---

### Step 2: Eliminate \(y\)
Subtract equation (2) from equation (1):
\[
(-4x + y) - (-x + y) = -29 - (-2)
\]
Simplify:
\[
-4x + y + x - y = -29 + 2
\]
\[
-3x = -27
\]
Solve for \(x\):
\[
x = 9
\]

---

### Step 3: Substitute \(x = 9\) into one of the original equations
Using equation (2):  
\[
-x + y = -2
\]
Substitute \(x = 9\):  
\[
-(9) + y = -2
\]
\[
-9 + y = -2
\]
\[
y = 7
\]

---

### Final Solution:
\[
x = 9, \; y = 7
\]

---

Would you like a graphical explanation or details on a different method?  

### Related Questions:
1. How can we verify solutions to systems of equations?
2. What is the difference between substitution and elimination methods?
3. How do systems of equations work graphically?
4. Can this system of equations have infinitely many solutions?
5. What is the process for solving systems of three equations?

**Tip:** Always check your solution by substituting back into the original equations to ensure accuracy.

Discover how to solve the system of equations -4x + y = -29 and -x + y = -2 using the elimination method. This step-by-step guide covers key algebraic techniques and provides a clear solution: x = 9, y = 7. Perfect for students in Grades 8-10 looking to strengthen their understanding of linear equations.

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