Problem 68 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(4(6 y+9)+7\)

Step-by-Step Solution

Answer

The simplified expression is 24y + 43.

Step 1: Distribute the constant

Multiply 4 by each term inside the parentheses: 4(6y + 9) becomes 4 * 6y + 4 * 9

Step 2: Simplify the products

Calculate the products: 4 * 6y = 24y and 4 * 9 = 36 So the expression becomes 24y + 36

Step 3: Add the constant term

Finally, add the constant 7 to the simplified expression: 24y + 36 + 7 Combine the constant terms: 36 + 7 = 43 Thus, the expression simplifies to 24y + 43.

Key Concepts

Distributive Property

The distributive property is an essential tool in algebra. It allows us to simplify expressions where a single term is multiplied by a group of terms within parentheses. In our example, we are given the expression \[4(6y + 9) + 7.\]
Using the distributive property, we multiply the number outside the parentheses (4) by each term inside the parentheses (6y and 9). Here's how it's done:

• First, multiply: \[4 \times 6y = 24y.\]
• Next, multiply: \[4 \times 9 = 36.\]

Now we have transformed the expression into: \[24y + 36.\]When applied correctly, the distributive property helps break down more complex expressions into simpler parts, making them easier to manage.

Combining Like Terms

Combining like terms is a crucial step in the simplification process. It involves merging terms that have the same variable components. In our simplified expression,\[24y + 36,\]there is only one 'y' term and no other 'y' terms to combine with. However, we still have another step where we need to add a constant (7) to the existing constant (36):\[24y + 36 + 7.\]
Here, both 36 and 7 are constant terms. We can add them together to further simplify our expression:

• Calculate: \[36 + 7 = 43.\]

Now, the expression becomes:\[24y + 43.\]By combining like terms, we ensure our expression is as simple as possible, making it easier to understand and work with.

Algebraic Simplification

Algebraic simplification means reducing an expression to its simplest form. This process involves applying several algebraic rules, such as the distributive property and combining like terms, to condense the expression as much as possible. Let's take our example through this journey: \[4(6y + 9) + 7.\]
First, we use the distributive property to expand the expression:\[4 \times 6y + 4 \times 9 = 24y + 36.\]
Next, we add the constant term 7:\[24y + 36 + 7.\]
Finally, we combine the constant terms 36 and 7:\[24y + 43.\]
The result, \[24y + 43,\]is the simplified form of the original expression. Simplifying algebraic expressions not only helps in solving equations but also allows for a clearer understanding of the relationships between different terms.