Answer:To prove ATVY = AXWY, we can use the given statements:
Given: TY = YX, ZTVY = ZXWY
We need to show that ATVY = AXWY.
Here is a step-by-step proof:
1. Start with the given statement ZTVY = ZXWY.
Reason: Given.
2. From statement 1, we can isolate TY by subtracting Z from both sides:
TVY = XWY.
Reason: Subtraction property of equality.
3. Now, let's use the given statement TY = YX.
Reason: Given.
4. Substitute TY with YX in statement 3:
YXV = XWY.
Reason: Substitution property of equality.
5. Rearrange statement 4 to match the desired expression.
XAVY = XWY.
Reason: Commutative property of multiplication.
6. Cancel out the common term X from both sides in statement 5:
AVY = WY.
Reason: Division property of equality.
7. Finally, replace AVY with ATVY and WY with AXWY:
ATVY = AXWY.
Reason: Substitution property of equality.
Therefore, we have successfully proven that ATVY = AXWY based on the given statements.