In algebraic simplification, find the expression that must be added to 8(3x - 4y) so that the final result becomes exactly 18x - 18y.

Introduction / Context:
This question tests your ability to manipulate algebraic expressions and match coefficients. You are given a scaled binomial 8(3x − 4y) and asked what must be added to it to obtain the target expression 18x − 18y. This is similar to balancing equations and is a useful technique in simplifying expressions and solving linear algebra problems in aptitude tests.

Given Data / Assumptions:

• Starting expression: 8(3x − 4y).
• Final expression required: 18x − 18y.
• We must find an expression E such that 8(3x − 4y) + E = 18x − 18y.
• x and y are variables representing real numbers.

Concept / Approach:
First, expand 8(3x − 4y) to obtain a simpler form. Then set up an equation where the unknown expression E bridges the difference between this expanded form and the desired final expression. By subtracting the known expression from the target, we derive E directly. Finally, we compare E with the given options and select the matching one. Matching coefficients of x and y ensures correctness.

Step-by-Step Solution:
1) Expand the given expression: 8(3x − 4y) = 24x − 32y. 2) We want 8(3x − 4y) + E = 18x − 18y, so substitute the expansion: 24x − 32y + E = 18x − 18y. 3) Solve for E by subtracting 24x − 32y from both sides: E = (18x − 18y) − (24x − 32y). 4) Distribute the negative sign: E = 18x − 18y − 24x + 32y. 5) Combine like terms: (18x − 24x) = −6x and (−18y + 32y) = 14y. 6) Therefore, E = −6x + 14y.

Verification / Alternative check:
Check by substitution: 8(3x − 4y) + (−6x + 14y) = 24x − 32y − 6x + 14y. Combine like terms to get (24x − 6x) + (−32y + 14y) = 18x − 18y, which matches the target expression exactly. This confirms that adding −6x + 14y to 8(3x − 4y) indeed produces 18x − 18y without any discrepancy.

Why Other Options Are Wrong:
Option a (6x − 14y) would give 24x − 32y + 6x − 14y = 30x − 46y, which does not match the required coefficients. Option b (14y + 6x) is the same as 6x + 14y and leads to the same incorrect result. Option c (14y − 6x) rearranges signs incorrectly and does not match E. Option d (6xy) introduces a new term xy, which is not present in the original or target expressions. Only option e, −6x + 14y, correctly balances the equation.

Common Pitfalls:
Students sometimes expand incorrectly, for example writing 8(3x − 4y) as 24x − 4y or mixing up the signs when subtracting expressions. Another mistake is to guess by inspection without actually matching coefficients, which can be misleading when several options look similar. Always expand completely and carefully subtract to find the exact expression required.

Final Answer:
The expression that must be added is -6x + 14y.