What expression should be added to 3(x − 2y) in order to obtain 2(3x + y) − 5(2x + 3)?

Introduction / Context:
This question is an algebraic manipulation problem that asks you to find an expression which, when added to a given expression, produces a specified result. It tests your understanding of expanding brackets, combining like terms, and rearranging algebraic expressions.

Given Data / Assumptions:

- Starting expression: 3(x − 2y)
- Target expression: 2(3x + y) − 5(2x + 3)
- We seek an expression E such that 3(x − 2y) + E = 2(3x + y) − 5(2x + 3)

Concept / Approach:
The key idea is to expand both sides and isolate E. First expand 3(x − 2y). Then expand 2(3x + y) and −5(2x + 3) and combine them into a single simplified expression. Once the right hand side is simplified, subtract 3(x − 2y) from it to obtain E, and simplify that result.

Step-by-Step Solution:
Step 1: Expand 3(x − 2y). This gives 3x − 6y.Step 2: Expand 2(3x + y). This gives 6x + 2y.Step 3: Expand −5(2x + 3). This gives −10x − 15.Step 4: Combine the results from Steps 2 and 3 to simplify the target expression: (6x + 2y) + (−10x − 15) = (6x − 10x) + 2y − 15 = −4x + 2y − 15.Step 5: We need E such that 3x − 6y + E = −4x + 2y − 15. Therefore E = (−4x + 2y − 15) − (3x − 6y).Step 6: Simplify E by distributing the minus sign: E = −4x + 2y − 15 − 3x + 6y = (−4x − 3x) + (2y + 6y) − 15 = −7x + 8y − 15.Step 7: Write E in a standard format, for example 8y − 7x − 15.

Verification / Alternative check:
Substitute E = 8y − 7x − 15 back into the original relationship: 3(x − 2y) + (8y − 7x − 15). This equals 3x − 6y − 7x + 8y − 15 = (3x − 7x) + (−6y + 8y) − 15 = −4x + 2y − 15. This matches the previously simplified form of 2(3x + y) − 5(2x + 3), confirming that E is correct.

Why Other Options Are Wrong:
- 8y − 7x + 15 changes the sign of the constant term and does not yield the correct target expression.
- 8y + 7x + 15 and 8y + 7x − 15 both have a positive 7x term instead of −7x, which leads to a mismatch when simplifying.
- 7x − 8y − 15 reverses the roles of the x and y coefficients and does not correct the expression to the required target.

Common Pitfalls:
Students may distribute the negative sign incorrectly when subtracting the expression 3(x − 2y) or make sign errors during expansion. It is also easy to forget to combine like terms fully. Writing each expansion step clearly and checking each coefficient carefully helps avoid these errors.

Final Answer:
The expression to be added is 8y − 7x − 15.