Variables and Algebraic Expressions: Comparison to an existing expression

To solve this problem, we will simplify each expression given in options a through f, and compare each to the expression $9a + 5n$.

• a. $4(a+n)+5a\cdot n$
  Expanding, we have:
  $4a + 4n + 5an$.
  The presence of the term $5an$ makes it different from $9a + 5n$. Therefore, it is not equivalent.
• b. $3(a-n)+6a+8n$
  Expanding, we have:
  $3a - 3n + 6a + 8n$.
  Combining like terms, we get:
  $(3a + 6a) + (-3n + 8n) = 9a + 5n$.
  This matches $9a + 5n$.
• c. $20a-7n-11a+12n$
  Combining like terms, we have:
  $(20a - 11a) + (-7n + 12n) = 9a + 5n$.
  This also matches $9a + 5n$.
• d. $10a-n+6a-n$
  Combining like terms, we have:
  $(10a + 6a) + (-n - n) = 16a - 2n$.
  This is not equivalent to $9a + 5n$.
• e. $9(a+n)-\frac{4n}{2}$
  Expanding and simplifying, we have:
  $9a + 9n - 2n = 9a + 7n$.
  This is not $9a + 5n$.
• f. $3(a+n)(n-5)+24a-3n^2-3an+20n$
  Expanding $3(a+n)(n-5)$, we have:
  $3(an + n^2 - 5a - 5n) = 3an + 3n^2 - 15a - 15n$.
  The expression becomes:
  $(3an + 3n^2 - 15a - 15n) + 24a - 3n^2 - 3an + 20n$.
  Combining like terms, we find:
  $9a + 5n$.
  This matches $9a + 5n$.

Therefore, the expressions that are equal to $9a + 5n$ are options b, c, and f.

The correct answer choice is b, c, f.

Let's simplify each expression option and compare them with the original expression $9y + 4x + 35$:

a. $3y + 2x + 6y + 30 + 7$

• Combine like terms:
  $3y + 6y = 9y$ and $2x$, $30 + 7 = 37$.
• The expression becomes: $9y + 2x + 37$, which is not equivalent to the original.

b. $5y + 3x + 4y + 30 + x + 5$

• Combine like terms:
  $5y + 4y = 9y$, $3x + x = 4x$, $30 + 5 = 35$.
• The expression becomes: $9y + 4x + 35$, which matches the original expression.

c. $4x + 30 + 9y + 5$

• Rearrange and combine constants:
  $4x + 9y + (30 + 5 = 35)$.
• The expression becomes: $9y + 4x + 35$, which matches the original expression.

d. $3(3y + 10) + 5 + 2 \cdot 2x$

• Distribute and simplify:
  $3 \cdot 3y = 9y$, $3 \cdot 10 = 30$, $5$, $2 \cdot 2x = 4x$.
• The expression becomes: $9y + 30 + 5 + 4x = 9y + 4x + 35$, which matches the original expression.

e. $5(8 + x) - x + 3y \cdot 3y - 5$

• Distribute and simplify:
  $5 \cdot 8 = 40$, $5 \cdot x = 5x$, $-x$, and $3y \cdot 3y = 9y^2$.
• The expression becomes: $40 + 5x - x + 9y^2 - 5$.
• Arrange: $4x + 9y^2 + 35$, which does not match the original expression.

f. $2x \cdot 2 + 3(2y + 10) + 5 + 3y$

• Distribute and simplify:
  $2x \cdot 2 = 4x$, $3 \cdot 2y = 6y$, $3 \cdot 10 = 30$, $5$, and $3y$.
• The expression becomes: $4x + 6y + 3y + 30 + 5$.
• Combine like terms:
  $6y + 3y = 9y$.
• The expression simplifies to $9y + 4x + 35$, which matches the original expression.

Therefore, the expressions a, e do not match the original, while b, c, d, and f do, confirming these are equivalent to $9y + 4x + 35$.

The correct answer is b. c. d. f.

To solve this problem, we will simplify each given expression and compare the results to the initial expression $45 + 5a + 3ab + 27b$.

Let's analyze each option:

• Option a: $(5+3b)(a+9)$
  Distribute: $= 5a + 45 + 3ba + 27b$
  Which rearranges to: $45 + 5a + 3ab + 27b$
  This matches the original expression.
• Option b: $45+\frac{2}{3}a+4\frac{1}{3}ab+(3a+27)b$
  Simplify: $= 45 + \frac{2}{3}a + \frac{13}{3}ab + 3ab + 27b$
  The terms do not simplify to match the original expression $45 + 5a + 3ab + 27b$.
• Option c: $2ab+5(9+a)+ab+\frac{9}{a}\cdot\frac{3ba}{1}$
  Simplify: Complete multiplication and distribution: $= 2ab + 45 + 5a + ab + 27b$
  Combine like terms: $= 45 + 5a + 3ab + 27b$
  This matches the original expression.
• Option d: $45+8ab+27b$
  This expression: $= 45 + 8ab + 27b$
  Depends on the situation with $b$, might match if conditions hold, but generally does not match unless specified.

After careful comparison, option a and c definitely represent the same value as the original expression, while option d depends on specific conditions regarding $b$.

The correct answer is: a, c, d (d. depends on b).