What expression must be added to 5(2x − y) so that the result becomes 4(2x − 3y) + 5(x + 4y)?

Introduction / Context:
This algebra question asks you to find an expression that must be added to one linear combination of x and y to obtain another. It tests your understanding of algebraic expansion, combining like terms, and solving simple equations with expressions. Such manipulations are common in simplifying linear expressions and in forming equivalent equations.

Given Data / Assumptions:
- We start with the expression 5(2x − y).
- We want to add an unknown expression E so that 5(2x − y) + E = 4(2x − 3y) + 5(x + 4y).
- x and y are real variables.
- We must solve for E and match it with one of the answer choices.

Concept / Approach:
The equation 5(2x − y) + E = 4(2x − 3y) + 5(x + 4y) can be rearranged as E = [4(2x − 3y) + 5(x + 4y)] − 5(2x − y). Hence, we simply expand all the brackets on the right, combine like terms, and the resulting simplified expression is E. This is a direct application of basic algebraic operations.

Step-by-Step Solution:
Step 1: Let E be the expression that needs to be added. Then 5(2x − y) + E = 4(2x − 3y) + 5(x + 4y). Step 2: Rearrange to isolate E: E = 4(2x − 3y) + 5(x + 4y) − 5(2x − y). Step 3: Expand 4(2x − 3y) to get 8x − 12y. Step 4: Expand 5(x + 4y) to get 5x + 20y. Step 5: Expand 5(2x − y) to get 10x − 5y. Step 6: Substitute these expansions into the expression for E to get E = (8x − 12y) + (5x + 20y) − (10x − 5y). Step 7: Combine like terms in stages. First add (8x − 12y) and (5x + 20y): this gives (8x + 5x) + (−12y + 20y) = 13x + 8y. Step 8: Now subtract (10x − 5y): E = (13x + 8y) − (10x − 5y) = 13x + 8y − 10x + 5y. Step 9: Combine the final like terms: (13x − 10x) + (8y + 5y) = 3x + 13y. Step 10: Therefore, the required expression E is 3x + 13y.
Verification / Alternative check:
To verify, plug E = 3x + 13y back into the original equation. Compute 5(2x − y) + E = (10x − 5y) + (3x + 13y) = 13x + 8y. Now evaluate the right hand side: 4(2x − 3y) + 5(x + 4y) = (8x − 12y) + (5x + 20y) = 13x + 8y. Since both sides give 13x + 8y, the equation holds, confirming that E is correct.

Why Other Options Are Wrong:
Option a, 3x − 13y, would lead to 10x − 5y + (3x − 13y) = 13x − 18y, which does not match 13x + 8y.
Option c, 13x − 3y, and option d, 13x + 3y, produce different coefficients for x and y that do not align with the right hand side expression after expansion.
Option e, x + 3y, is too small in both x and y coefficients to make the expressions equal.

Common Pitfalls:
Students sometimes mis distribute the negative sign when subtracting the expression 5(2x − y), writing −5(2x − y) incorrectly as −5(2x) − y instead of −10x + 5y. Another common error is mixing up coefficients when combining like terms. Writing each step clearly, particularly the expansion and the subtraction, helps avoid such mistakes. Always verify by substituting back into the original equation to confirm equality.

Final Answer:
The expression that must be added is 3x + 13y.