To solve the system of equations by elimination, we proceed as follows:

Given system:

1. $-x + y = -1$
2. $4x - y = 4$

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Step 1: Add the two equations together

$(-x + y) + (4x - y) = -1 + 4$ Simplify: $- x + 4x + y - y = 3$ $3x = 3$

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Step 2: Solve for $x$

$x = \frac{3}{3} = 1$

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Step 3: Substitute $x = 1$ into one of the original equations

Substitute into $-x + y = -1$: $-(1) + y = -1$ $-1 + y = -1$ $y = 0$

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Final Solution:

$x = 1, \, y = 0$

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Verify the solution:

Substitute $x = 1$ and $y = 0$ into both equations:

1. $-x + y = -1$: $-1 + 0 = -1$ ✅
2. $4x - y = 4$: $4(1) - 0 = 4$ ✅

The solution is correct: $(x, y) = (1, 0)$.

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Would you like further details or clarifications?

Related Questions:

1. How do you determine whether elimination or substitution is the best method?
2. What are some examples where elimination fails or requires additional steps?
3. Can you graphically interpret the solution $(1, 0)$ for this system?
4. What happens if the two equations are parallel?
5. How do you modify the elimination method for three-variable systems?

Tip: When using elimination, always ensure terms align properly before combining equations to simplify calculations.