Problem 60 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ 11 x+3(8-x) $$

Step-by-Step Solution

Answer

The simplification of the expression \(11x + 3(8 - x)\) is \(8x + 24\).

Step 1: Distribute multiplication

Multiply 3 by each term in the brackets separately, resulting in \(11x + 24 - 3x\).

Step 2: Combine like terms

Combine together \(11x\) and \(-3x\) to simplify the equation, resulting in \(8x + 24\).

Key Concepts

Distributive Property

The Distributive Property is a fundamental concept in algebra that involves multiplying a single term across terms within parentheses. This property allows you to eliminate parentheses by spreading, or "distributing," the multiplication over each term inside. For example, in the expression \(3(8-x)\), you apply the distributive property by multiplying 3 with each term inside the brackets. This gives you \(3 \times 8\) and \(3 \times (-x)\).

• The result for the first part is \(24\), since \(3 \times 8 = 24\).
• The second part is \(-3x\), since \(3 \times (-x) = -3x\).

Now, the entire part inside the parentheses is transformed from \(3(8 - x)\) into \(24 - 3x\). The expression, when simplified further, becomes \(11x + 24 - 3x\). Using the distributive property effectively helps to simplify expressions and prepare them for further operations like combining like terms.

Combining Like Terms

Combining like terms is the process of simplifying an algebraic expression by merging terms that have the same variable parts and exponents. This step is crucial as it helps reduce expressions to their simplest forms. Let's continue with the expression obtained after distributing: \(11x + 24 - 3x\).

• Identify terms with the same variable, which are \(11x\) and \(-3x\).
• "Combine" them by adding or subtracting their coefficients. In this case, arrange \(11x - 3x\).
• This operation gives \((11 - 3)x = 8x\).

As you can see, combining like terms simplifies the expression to \(8x + 24\). It helps make complex equations more manageable, paving the way towards solution or further simplification.

Simplification

Simplification in algebra is the process of reducing an expression to its most concise form without changing its value. As we've solved previously, the expression \(11x + 3(8-x)\) was first transformed using the distributive property and then further simplified by combining like terms to become \(8x + 24\).

The steps to ensure full simplification are:- **Apply operations** like distribution and combining like terms.- **Ensure no redundancies** such as repeated terms or unnecessary parentheses remain.- **Double-check** arithmetic operations, such as addition or subtraction.
What simplification does is streamlining the expression, making it easier to interpret or solve, and reducing potential calculation errors.

Linear Expressions

Linear expressions are algebraic expressions where each term is either a constant or the product of a constant and a single variable, raised to the power of one (or zero, for just constant terms). They are called "linear" because they map to straight lines when graphed.

For example, in the simplified expression \(8x + 24\), we have:- **\(8x\)** as the variable term with a coefficient of 8.- **\(24\)** as the constant term.Linear expressions are straightforward because they do not involve variables raised to any power other than one and do not include any other operations like division by a variable, exponents, or roots. This simplicity makes them easier to solve and understand compared to quadratic or complex polynomial expressions.
Recognizing and working with linear expressions is fundamental in algebra as they are often the first type of expressions one deals with when learning about functions and equations.