Algebra I / Chapter 4: Fractions / Fractions
Proof: Remove Common Term
Let's prove the following theorem:
if the following are true:
- a ⋅ x = a ⋅ y
- not (a = 0)
then x = y
Proof:
Given
1 | a ⋅ x = a ⋅ y |
---|---|
2 | not (a = 0) |
# | Claim | Reason |
---|---|---|
1 | (1 / a) ⋅ (a ⋅ x) = (1 / a) ⋅ (a ⋅ y) | if a ⋅ x = a ⋅ y, then (1 / a) ⋅ (a ⋅ x) = (1 / a) ⋅ (a ⋅ y) |
2 | (1 / a) ⋅ (a ⋅ x) = x | if not (a = 0), then (1 / a) ⋅ (a ⋅ x) = x |
3 | (1 / a) ⋅ (a ⋅ y) = y | if not (a = 0), then (1 / a) ⋅ (a ⋅ y) = y |
4 | x = y | if (1 / a) ⋅ (a ⋅ x) = (1 / a) ⋅ (a ⋅ y) and (1 / a) ⋅ (a ⋅ x) = x and (1 / a) ⋅ (a ⋅ y) = y, then x = y |
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