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Chapter 5: Problem 4

Write an equivalent expression for: $$ 11(x+4)= $$ $$ 10 x-x+4 x= $$

Short Answer

Expert verified
The simplified equivalent expressions are: \(11x + 44\) and \(13x\).

Step by step solution

01

Apply the Distributive Property

Distribute 11 through the parentheses, multiplying 11 by x and 4. \[11(x+4) = 11x + 44\] The simplified equivalent expression for the first expression is: \(11x + 44\) For the second expression: \(10x - x + 4x\)
02

Combine Like Terms

Combine the terms containing x. \[10x - x + 4x = (10 - 1 + 4)x = 13x\] The simplified equivalent expression for the second expression is: \(13x\) The two equivalent expressions are: \[11x + 44\] \[13x\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a fundamental rule in algebra that facilitates the multiplication of a single term across each element within a bracket, thus allowing the simplification of expressions. It is expressed as, \(a(b + c) = ab + ac\). In the context of the exercise at hand the number 11 is distributed across both \(x\) and 4 within the brackets: \(11(x + 4) = 11x + 44\). This is instrumental in breaking down more complex problems into manageable calculations, allowing us to merge constants and like terms with greater ease.
Combine Like Terms
To combine like terms is to simplify an algebraic expression by adding or subtracting terms that have the same variables raised to the same power. In \(10x - x + 4x\), we are only working with one variable \(x\), but we can combine the coefficients (10, -1, and 4) that precede it. By performing the necessary addition and subtraction (\(10 - 1 + 4\)) we get \(13x\). This technique tidies up an expression, streamlines calculations, and leads to a more compact form which is much easier to interpret and use in subsequent mathematical operations.
Simplifying Expressions
Simplifying expressions involves reducing an algebraic expression to its simplest form. This process includes expanding brackets using the distributive property and then combining like terms. For example, after using the distributive property to expand \(11(x + 4)\), we simplify to get \(11x + 44\). The idea is to transform the expression so that it has no parentheses, no like terms, and is as brief as possible without changing the expression's value. Simplifying helps in various mathematical procedures, and it often makes other operations, like solving equations, more straightforward.
Mathematical Operations
Mathematical operations such as addition, subtraction, multiplication, and division are the building blocks of algebra. In our exercise, we used mathematical operations to distribute the 11 into the parenthetical term (multiplication), and then to combine like terms by adding and subtracting coefficients of \(x\). Understanding how to accurately manipulate these operations is crucial to solve algebraic expressions efficiently and forms the backbone of most mathematical tasks.

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