Problem 57 - Algebra A Combined Function

Question

Use the distributive property to write each expression without parentheses. Then simplify the result, if possible. See Examples 7 through 12. $$ 3(2 r+5)-7 $$

Step-by-Step Solution

Answer

The expression simplifies to \( 6r + 8 \).

Step 1: Apply the Distributive Property

The distributive property states that for any numbers \( a \), \( b \), and \( c \), \( a(b + c) = ab + ac \). In our expression \( 3(2r + 5) \), \( a = 3 \), \( b = 2r \), and \( c = 5 \). Distribute \( 3 \) across the terms inside the parentheses: \[ 3(2r + 5) = 3 \cdot 2r + 3 \cdot 5 \].

Step 2: Perform Multiplication

Multiply the terms: \( 3 \cdot 2r = 6r \) and \( 3 \cdot 5 = 15 \). So, the expression now becomes \( 6r + 15 \).

Step 3: Simplify the Expression by Subtracting

Substitute the simplified expression from distributing back into the original expression: \( 6r + 15 - 7 \). Now, simplify by combining like terms. Subtract \( 7 \) from \( 15 \) to get \( 15 - 7 = 8 \). So the expression becomes \( 6r + 8 \).

Key Concepts

Understanding Algebra

Algebra forms the basis of numerous mathematical problems and solutions. It involves working with symbols and numbers to solve for unknowns, often using a variety of operations.
The main goal is to find the unknown values that satisfy given equations or expressions. In our exercise, algebra helps us manipulate the expression using set rules like the distributive property. By applying algebraic rules, such as multiplication and subtraction, we aim to simplify complex expressions into more manageable forms.
In short, algebra is like a puzzle, where you use given clues (the rules and operations) to solve for the unknown or simplify an expression.

Simplifying Expressions

Simplifying expressions is a key skill in algebra that involves making an expression as concise as possible. In this process, you perform operations to reduce the expression to its most straightforward form without changing its meaning or value.
Take, for instance, our expression: after distributing and multiplying, we reach a form like \(6r + 15\). To simplify further, we apply additional operations like subtraction, giving us a final expression of \(6r + 8\).
This process uses several steps:

• Distributing coefficients across terms
• Performing multiplication or division
• Adding or subtracting terms

These steps aim to combine the components of the expression into a more straightforward form, without any unnecessary complexity.

Combining Like Terms

Combining like terms is a fundamental concept in simplifying expressions. 'Like terms' refer to terms in an expression that have the same variable raised to the same power. This means that only the coefficients (the numbers in front of the variables) need to be combined.
For our exercise, after applying the distributive property and simplifying the multiplication, we looked for like terms to combine. We had the expression \(6r + 15 - 7\). Here, \(15\) and \(7\) are like terms since both are numbers without variables.
By subtracting \(7\) from \(15\), we combined these like terms, which simplified the expression to \(6r + 8\).
This final step of combining like terms is crucial for ensuring the expression is fully simplified and free of redundancy, making it easier to interpret and use in further calculations.