Variables and Algebraic Expressions: Comparison to an existing expression

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 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.

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).