To find the equation of the translated line, we first start with the original line equation:
\[ y = -x + 1. \]
Now, we will translate this line 2 units to the right and 2 units up.
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Translation to the right (2 units):
- When we translate a graph to the right, we replace \(x\) with \(x - a\) where \(a\) is the number of units we move to the right. Here, \(a = 2\).
- So, we replace \(x\) in the original equation: \[ y = - (x - 2) + 1. \] This simplifies to: \[ y = -x + 2 + 1 = -x + 3. \]
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Translation upwards (2 units):
- When moving the graph up by \(b\) units, we add \(b\) to the equation. Here, \(b = 2\).
- Adding 2 to our equation: \[ y = -x + 3 + 2 = -x + 5. \]
Thus, the equation of the translated line is:
\[ \boxed{y = -x + 5}. \]