Problem 33 - Algebra A Combined Function

Question

Remove parentheses and simplify each expression. $$ 7(d-3)+10 $$

Step-by-Step Solution

Answer

The simplified expression is \(7d - 11\).

Step 1: Distribute the 7 into the Parentheses

The first step is to remove the parentheses by distributing the 7 across the terms inside the parentheses. For the expression \(7(d-3)+10\), apply the distributive property: \(7 \times d\) and \(7 \times (-3)\). This gives the expression \(7d - 21\).

Step 2: Combine Like Terms

Now substitute the expanded terms back into the expression, giving \(7d - 21 + 10\). Combine the constant terms \(-21\) and \(+10\) by adding them together: \(-21 + 10 = -11\). Thus, the simplified expression is \(7d - 11\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps simplify expressions and solve equations. It allows you to "distribute" a single term across terms inside parentheses. In our example, we start with the expression \(7(d-3)+10\). This means we will distribute the 7 to both \(d\) and \(-3\).

• First, multiply 7 by \(d\): this gives \(7d\).
• Next, multiply 7 by \(-3\): this gives \(-21\).

After applying the distributive property, the expression inside the parentheses is replaced with \(7d - 21\). This step is crucial as it prepares the expression for further simplification by isolating each variable and constant separately.

Combining Like Terms

After using the distributive property, the next step is to simplify the expression by combining like terms. This process helps to consolidate terms that share the same variable or are constants into a single term.

In our exercise, we have the expression \(7d - 21 + 10\) after removing the parentheses. Notice that \(-21\) and \(+10\) are both constant terms:

• Add these constants together: \(-21 + 10\).
• This equals \(-11\).

So, combining them further simplifies the expression to \(7d - 11\). By combining like terms, you reduce the number of terms and simplify the algebraic expression, making it easier to work with.

Removing Parentheses

Removing parentheses is a key procedure in simplifying algebraic expressions. Parentheses often group terms together, dictating the order of operations in an expression. By removing them correctly, you can simplify expressions and solve equations easier.

In this exercise, parentheses are dealt with using the distributive property, where we distribute the multiplier (7) outside the parentheses to each term inside: \(7 \times d\) and \(7 \times (-3)\). This eliminates the parentheses and sets the stage for further simplification.

• Ensure every term inside the parentheses is accounted for in the distribution step.
• After removing the parentheses, assess the expression for any like terms that can be combined.

Through such procedures, removing parentheses becomes a manageable and systematic task, allowing you to focus on simplifying the broader expression efficiently.