Problem 41 - Algebra 2

Question

Simplify each expression. $$ 4(14 c-10 d)-6(d+4 c) $$

Step-by-Step Solution

Answer

The simplified expression is \(32c - 46d\).

Step 1: Distribute the coefficients

We start by distributing the numbers outside the parentheses with each term inside the parentheses. This means multiplying every term inside the parentheses by the number outside. For the first term, distribute 4 into both expressions inside the parentheses: \(4 \times 14c = 56c\) and \(4 \times (-10d) = -40d\). For the second expression, distribute -6 into both terms: \(-6 \times d = -6d\) and \(-6 \times 4c = -24c\). This gives us: \[ 56c - 40d - 6d - 24c \]

Step 2: Combine like terms

Now, we need to simplify the expression by combining like terms. The like terms here are the terms with \(c\) and the terms with \(d\). Combine the \(c\) terms: \(56c - 24c = 32c\). Combine the \(d\) terms: \(-40d - 6d = -46d\). This results in the simplified expression: \[ 32c - 46d \]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by each term within a parenthesis. It simplifies expressions and helps us solve equations more easily. For example, if you have a term outside the parenthesis like 4 in our problem, you distribute it across every term inside the parenthesis.

• The first step is to take the term outside the parentheses, which is 4, and multiply it with each term inside: 14c and -10d.
• This gives us two separate products: \(4 \times 14c = 56c\) and \(4 \times (-10d) = -40d\).

After repeating this process for the second part of the expression with -6 and distributing it, you achieve distributive results that lay the foundation for further simplification.

Like Terms

In algebra, 'like terms' are essential because they allow us to combine terms to make expressions easier to work with. Like terms are terms that have the same variable raised to the same power. In our expression, this means identifying terms that contain the same variable, like the 'c' terms or the 'd' terms.

• Our original distributed expression gives us these terms: 56c, -24c and -40d, -6d.
• You notice that 56c and -24c both contain the variable 'c' and can be combined.
• For the 'd' terms, -40d and -6d are similar, so they can be combined as well.

Combining these like terms helps us shrink the expression into a simpler form, making it much easier to handle.

Simplification

Simplification in algebra involves reducing expressions to their simplest form. It's crucial for understanding and solving algebraic expressions efficiently. After applying the distributive property and identifying like terms, the next task is simplification.

• Start by grouping like terms: combine 56c and -24c to get 32c.
• Next, combine the 'd' terms: -40d and -6d sum up to -46d.

As a result, your fully simplified expression becomes \(32c - 46d\). This form is much less complex and offers quick insight into the relationship between the variables involved.