Question

Remove parentheses and simplify. $$(c-5)-(6+3 c)$$

Short answer

\( -2c - 11\)

Step-by-step solution

Distribute the Negative Sign

Apply the distributive property to remove the parentheses. Multiply the negative sign outside the second parentheses by each term inside the parentheses: \(c - 5 - (6 + 3c) = c - 5 - 6 - 3c\).

Combine Like Terms

Combine the like terms by adding or subtracting the coefficients: \(c - 5 - 6 - 3c = c - 3c - 5 - 6\).

Simplify the Expression

Finish combining the like terms to simplify the expression: \(c - 3c = -2c\) and \( -5 - 6 = -11\), resulting in \( -2c - 11\).

Key concepts

Distributive Property

Understanding the distributive property is like acquiring a master key for unlocking algebraic expressions. It's a rule that lets you multiply a single term by each term within a set of parentheses.
Consider an expression like \(a(b + c)\). Instead of adding \(b + c\) first, the distributive property allows you to multiply \(a\) with \(b\) and \(a\) with \(c\) separately. In essence, you’re 'distributing' \(a\) over the addition within the parentheses, resulting in \(ab + ac\).
In the exercise at hand, a negative sign acts as a multiplier for the terms inside the parentheses. Applying our rule, we distribute this sign to \(6\) and \(3c\), flipping their signs to yield \(c - 5 - 6 - 3c\). Always remember that a negative sign in front of the parentheses is like multiplying by -1.

Combine Like Terms

Moving forward to the idea of gathering 'like terms', imagine you have a basket of fruits where you group apples with apples and oranges with oranges. Algebraically, like terms are those that have the exact same variables raised to the same powers.
In our example, the like terms are those with the variable \(c\) and those without any variable (known as constant terms). To combine them, you simply add or subtract their coefficients. For instance, \(c\) and \(3c\) have the same variable, so we combine them to get \(c - 3c = -2c\). Separately, we combine the constants \(5\) and \(6\) to get \( -5 - 6 = -11\).
Always keep an eye for these like terms as combining them is a fundamental step towards simplifying algebraic expressions.

Algebraic Simplification

Algebraic simplification is the art of reducing expressions to their most basic form. It's the process of making an algebraic expression as straightforward and concise as possible. In our exercise, after distributing the negative and combining like terms, we are left with \( -2c - 11\).
This expression now has no parentheses and each variable term and constant term are as simple as they can be. There's nothing more to combine, and the expression is in its most reduced form. Simplifying an algebraic expression not only makes it easier to understand but also prepares it for further operations such as solving equations or graphing.