Find the sum of the three algebraic expressions (4x + 3y - 7z), (x - y + z), and (2x + 3y + 4z). What is the resulting expression?

Introduction / Context:
This question involves adding three linear algebraic expressions in the variables x, y, and z. Simplifying such sums tests basic skills in combining like terms, which is a core algebra technique used in many topics including polynomials, equations, and vector expressions.

Given Data / Assumptions:

• First expression: 4x + 3y - 7z
• Second expression: x - y + z
• Third expression: 2x + 3y + 4z
• We must find their sum as a single simplified expression in x, y, and z.

Concept / Approach:
When adding algebraic expressions, we group like terms:

• Combine all x terms together.
• Combine all y terms together.
• Combine all z terms together.

The coefficients of each variable are added arithmetically, and the variables themselves remain the same.

Step-by-Step Solution:
Write the sum explicitly: (4x + 3y - 7z) + (x - y + z) + (2x + 3y + 4z) Group x terms: 4x + x + 2x = 7x Group y terms: 3y - y + 3y = 5y Group z terms: -7z + z + 4z Compute z coefficient: -7 + 1 + 4 = -2, so z term is -2z Therefore the sum is 7x + 5y - 2z

Verification / Alternative check:
Choose simple values such as x = 1, y = 1, z = 1 to verify. The first expression is 4 + 3 - 7 = 0. The second is 1 - 1 + 1 = 1. The third is 2 + 3 + 4 = 9. Total is 0 + 1 + 9 = 10. Now evaluate 7x + 5y - 2z at x = y = z = 1, which gives 7 + 5 - 2 = 10. The results match, confirming the simplification.

Why Other Options Are Wrong:
Option b, 7x + y - 2z, underestimates the coefficient of y. Option c, 3x + 5y - 2z, underestimates the coefficient of x. Option d, 7x + 5y + 2z, has the wrong sign on the z term. Option e, 3x - y + 4z, does not match any of the combined coefficients and clearly differs from the correct sum.

Common Pitfalls:
A common mistake is to incorrectly add coefficients, especially when negative signs are involved. Students may treat -7z + z as -8z instead of -6z before adding the remaining 4z. Writing each group of like terms in one line and carefully computing the sum of coefficients helps avoid such errors.

Final Answer:
The resulting expression after addition is 7x + 5y - 2z.