To solve the system of linear equations:  
\[
-x + y = 4
\]  
\[
x + 3y = 4
\]

### Step 1: Rearrange the equations for clarity:
1. \( -x + y = 4 \)
2. \( x + 3y = 4 \)

### Step 2: Eliminate one variable.
Add the two equations together to eliminate \(x\):  
\[
(-x + y) + (x + 3y) = 4 + 4
\]
\[
4y = 8
\]
\[
y = 2
\]

### Step 3: Solve for \(x\).
Substitute \(y = 2\) into either original equation. Using the second equation \(x + 3y = 4\):  
\[
x + 3(2) = 4
\]
\[
x + 6 = 4
\]
\[
x = -2
\]

### Final Solution:
\[
x = -2, \, y = 2
\]

Would you like a detailed explanation or verification?  

### Related Questions:
1. How do you graphically solve this system of equations?
2. What happens if you eliminate \(y\) first instead of \(x\)?
3. How can substitution be used to solve this system instead of elimination?
4. What are the properties of systems of equations with unique solutions?
5. Can this system be solved using matrices or determinants?

### Tip:
When solving linear systems, always double-check your solution by substituting the values back into the original equations.

Solve the system of equations: -x + y = 4 and x + 3y = 4

Learn how to solve the system of equations -x + y = 4 and x + 3y = 4 using the elimination method. This step-by-step solution demonstrates key algebraic techniques and verifies the solution x = -2, y = 2. Suitable for Grades 8-10.

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