Question

Combine as indicated and simplify. $$(4 x+2)+(3 x-4)-(5 x+5)$$

Short answer

The simplified form of the expression is \(2x - 7\).

Step-by-step solution

Expand the parentheses

Apply the distributive property to remove the parentheses. For the negative parentheses, change the sign of each term inside the parentheses when removing them. The expression becomes: \(4x + 2 + 3x - 4 - 5x - 5\).

Combine like terms

Combine the coefficients of the like terms, which are terms with the same variable part. Combine \(4x\), \(3x\), and \(-5x\) and also combine the constant terms \(2\), \(-4\), and \(-5\). The simplified terms are \(2x\) and \(-7\).

Write the simplified expression

After combining like terms, write down the simplified expression: \(2x - 7\).

Key concepts

Simplifying Expressions

Simplifying algebraic expressions is a fundamental skill in algebra that allows students to condense an expression into its simplest form. This process often involves combining like terms, which are terms that have the same variable raised to the same power.

For instance, in the exercise \(4x + 2 + 3x - 4 - 5x - 5)\), we identify like terms as those having the variable \(x\)) and those that are constants. Simplifying such an expression requires careful addition or subtraction of the coefficients of like terms.

Here, we combine the coefficients \(4, 3,\) and \( -5\)) for the terms with \(x\)), yielding \(2x\)), and we add \(2, -4,\) and \( -5\)) to get \( -7\)), the simplified constant term. The resulting expression, \(2x - 7\)), is much clearer and easier to work with for further algebraic operations.

Distributive Property

The distributive property is an essential algebraic property that enables us to multiply a single term across a sum or difference within parentheses. It is often signified as \(a(b + c) = ab + ac\)) or \(a(b - c) = ab - ac\)).

In the context of our exercise, the distributive property comes into play when dealing with the subtraction of the parentheses \( -(5x + 5) \)). We distribute the negative sign across each term inside the parentheses, transforming it into \( -5x - 5 \)).

Why Use the Distributive Property?

Using the distributive property ensures that each term inside the parentheses is correctly signed before combining like terms. It prevents errors that can occur from incorrectly removing parentheses, especially when negative signs are involved, as seen in our exercise. This property is a powerful tool that aids in accurately simplifying expressions.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. In algebra, the aim is to manipulate these expressions using established rules to solve for unknown variables or to simplify the expressions as much as possible.

In our given exercise, we have an algebraic expression comprised of variables (\textsc{x}), constants (\textsc{2, -4, -5}), and arithmetic operations (\textsc{+, -}). Through the process of simplifying this expression, we can make it more compact and understandable.

Understanding Variables and Constants

Variables, represented by letters like \(x\)) in our exercise, stand for unknown values that we often aim to solve for, while constants are fixed numbers. Grasping the role each component plays in an algebraic expression aids in performing accurate mathematical operations, which is particularly important when faced with complex or lengthy expressions.