Subtract $\frac{1}{2}$(a - b - c) from the sum of $\frac{1}{2}$(a - b + c) and $\frac{1}{2}$ (a + b - c)

Analyze the question: We add the first two expressions, then subtract ½(a − b − c) from that sum.

Concept: Take the common factor ½ out, then combine like terms. Be careful with the minus sign when subtracting a bracket: it changes every sign inside.

Solution:

Sum of the first two: ½(a − b + c) + ½(a + b − c) = ½[(a − b + c) + (a + b − c)] = ½[2a + 0b + 0c] = a

Now subtract ½(a − b − c):

a − ½(a − b − c) = a − ½a + ½b + ½c = ½a + ½b + ½c = ½(a + b + c)

Check with numbers: Let a = 3, b = 2, c = 1.

• ½(3 − 2 + 1) = 1
• ½(3 + 2 − 1) = 2
• Sum = 3
• ½(3 − 2 − 1) = 0
• 3 − 0 = 3
• ½(a + b + c) = ½(6) = 3 ✓.

Answer: A. ½(a + b + c)

Tip: When you subtract a bracket, flip every sign inside it. Also, "subtract X from Y" means Y − X, so the order matters. Substituting small numbers is a quick way to check your answer in the exam.