Match Equivalent Expressions: (x+7)(y+5) and Their Expanded Forms

To solve this problem, we'll match each bracketed pair of algebraic terms with its equivalent expanded form using the distributive property.

Step-by-Step Solution:

Expression 1: $(y+5)(x+7)$

• Apply FOIL method:
• First: $y \cdot x = xy$
  Outside: $y \cdot 7 = 7y$
  Inside: $5 \cdot x = 5x$
  Last: $5 \cdot 7 = 35$
• Combine: $xy + 7y + 5x + 35$
• Match: Option a $(xy + 7y + 5x + 35)$

Expression 2: $(x+5)(y+7)$

• Apply FOIL method:
• First: $x \cdot y = xy$
  Outside: $x \cdot 7 = 7x$
  Inside: $5 \cdot y = 5y$
  Last: $5 \cdot 7 = 35$
• Combine: $xy + 7x + 5y + 35$
• Match: Option b $(xy + 7x + 5y + 35)$

Expression 3: $(x-5)(y-7)$

• Apply FOIL method:
• First: $x \cdot y = xy$
  Outside: $x \cdot (-7) = -7x$
  Inside: $(-5) \cdot y = -5y$
  Last: $(-5) \cdot (-7) = 35$
• Combine: $xy - 7x - 5y + 35$
• Match: Option c $(xy - 7x - 5y + 35)$

Expression 4: $(x-5)(y+7)$

• Apply FOIL method:
• First: $x \cdot y = xy$
  Outside: $x \cdot 7 = 7x$
  Inside: $(-5) \cdot y = -5y$
  Last: $(-5) \cdot 7 = -35$
• Combine: $xy + 7x - 5y - 35$
• Match: Option d $(xy + 7x - 5y - 35)$

By matching each expression with its expanded equivalent, we conclude:

1-a, 2-b, 3-c, 4-d