Problem 16 - Elementary Algebra

Question

Use the distributive property to rewrite each of the following quantities. $$(x+6) 7$$

Step-by-Step Solution

Answer

Question: Rewrite the expression \((x+6)7\) using the distributive property. Answer: \(7x + 42\)

Step 1: Identify the factors and terms

In the expression \((x+6)7\), the factors are \((x+6)\) and \(7\). Inside the first factor, the terms are \(x\) and \(6\).

Step 2: Apply the distributive property

Apply the distributive property by multiplying the second factor (7) with each term inside the first factor (\((x+6)\)). $$7(x+6) = 7(x) + 7(6)$$

Step 3: Simplify the expression

Perform the multiplication in the expression: $$7x + 42$$ The rewritten expression using the distributive property is \(7x + 42\).

Key Concepts

Algebra

Algebra is a branch of mathematics that uses symbols, usually letters like \(x\), to represent numbers or quantities in formulas and equations. It allows us to formulate and solve problems in a general way, providing a powerful tool for reasoning and computation. In algebra, we often encounter expressions like \((x+6)7\), where our goal is to simplify or rewrite them using mathematical principles.

One of the key techniques used in algebra is the distributive property, which lets us expand expressions to make them easier to work with. This technique is useful for breaking down complex expressions into simpler components, making it a cornerstone skill in algebra.

Multiplication

Multiplication is a fundamental mathematical operation that involves adding a number to itself a specified number of times. It is one of the four basic arithmetic operations, alongside addition, subtraction, and division. When using multiplication in algebra, like in the expression \((x+6)7\), we "distribute" the multiplication across the terms inside the parentheses.

In the expression \((x+6)7\), we multiply each term inside the parentheses by the number outside. This means:

• Multiply \(x\) by 7 to get \(7x\).
• Multiply 6 by 7 to get 42.

These steps show how multiplication can be applied systematically to express numbers and variables in different forms, enabling easier computation and analysis.

Simplification

Simplification in mathematics means reducing expressions to their most basic form without changing their value. This often involves performing operations like addition, subtraction, multiplication, or division. Simplifying an expression helps to easily interpret and solve mathematical problems.

In the context of using the distributive property, once we've distributed and performed the multiplication, the next step is to simplify the expression. With an expression like \(7(x) + 7(6)\), we calculate:

• \(7x\) remains as it is unless further context is provided where it can be simplified.
• \(7 \times 6 = 42\)

Thus, the expression \(7(x) + 42\) is the simplified form of the original quantity.

Mathematical Expression

A mathematical expression is a combination of numbers, symbols, and operators (such as +, -, /, \(\times\)) that represents a value. Expressions can be as simple as a single number or letter, or as complex as a multi-step formula.

The expression \((x+6)7\) is a more complex expression initially because it involves a variable \(x\) and multiplication. Using principles like the distributive property, we can manipulate and transform these expressions.

• This expression combines addition inside the parentheses and multiplication outside.
• The goal is to simplify it into something like \(7x + 42\), which is easier to understand and work with.

Understanding how to construct and deconstruct mathematical expressions is critical in solving algebraic problems efficiently.