Q. Which Expression is Equivalent to \( 6(x + 2y) + 3 + 4y + 5 \)?

Answer

Distribute the 6.
\[ 6x + 12y + 3 + 4y + 5 \]
Combine like terms.
Combine the y terms and the constants.

\[ 6x + 16y + 8 \]
Factor (optional).
Factor out a 2.

\[ 2(3x + 8y + 4) \]

Detailed Explanation

Solution

Write the original expression exactly as given:

\(6(x + 2y) + 3 + 4y + 5\)
Apply the distributive property. The distributive property states that \(a(b + c) = ab + ac\). Multiply 6 by each term inside the parentheses:

\(6 \cdot x = 6x\)

\(6 \cdot 2y = 12y\)

After distribution the expression becomes:

\(6x + 12y + 3 + 4y + 5\)
Identify and group like terms. Like terms are terms with the same variable part. Here we have:
- x-terms: \(6x\)
- y-terms: \(12y\) and \(4y\)
- constant terms: \(3\) and \(5\)
Group them as:

\(6x + (12y + 4y) + (3 + 5)\)
Add the grouped like terms.

\(12y + 4y = 16y\)

\(3 + 5 = 8\)

So the expression simplifies to:

\(6x + 16y + 8\)
Final answer: \(6x + 16y + 8\)

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Frequently Asked Questions

What expression is equivalent to 6(x+2y)+3+4y+5?

Simplified: 6x + 16y + 8. Factored: 2(3x + 8y + 4).

How do you distribute 6 over (x+2y)?

Multiply 6 by each term: 6*x = 6x and 6*2y = 12y, giving 6x + 12y.

What does "combine like terms" mean here?

Add or subtract terms that have the same variable and exponent. Here 12y and 4y combine to 16y; 3 and 5 combine to 8.

Why can't I combine x and y terms?

x and y are different variables, so they represent different kinds of terms. Only terms with the same variable(s) and powers can be combined.

How can I check my work?

Reverse steps: expand any factored form to see if you get the original, or substitute random x and y value into both expressions to confirm equal results.

Can the final expression be factored further?

Yes: 6x + 16y + 8 has common factor 2, so 2(3x + 8y + 4). No further common factor among 3x, 8y, 4.

What common mistakes should I watch for?

Forgetting to distribute to both terms, combining unlike terms, or sign errors. Also remember to combine constant terms separately from variable terms.

Does the order of operations matter here?

Yes: distribute before combining. Parentheses indicate distribution; then perform additions by combining like terms.

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