Problem 68 - College Algebra

Question

What property of real numbers would simplify the following expression: \(4+7(x-1) ?\)

Step-by-Step Solution

Answer

Distributive property: \(7x - 3\).

Step 1: Identify the parts of the expression

The given expression is \(4 + 7(x-1)\). It contains a constant term \(4\) and a term involving the variable \(x\) within parentheses, which is \(7(x-1)\).

Step 2: Recall the distributive property

The distributive property states that for any numbers \(a\), \(b\), and \(c\), \(a(b + c) = ab + ac\). This property can be used to multiply a single term across terms inside parentheses.

Step 3: Apply the distributive property

Use the distributive property to simplify \(7(x - 1)\). This means multiplying \(7\) by both \(x\) and \(-1\), so \(7(x - 1) = 7x - 7\).

Step 4: Simplify the expression

Substitute the result of the distribution back into the original expression: \(4 + 7(x - 1)\) becomes \(4 + 7x - 7\).

Step 5: Combine like terms

Combine the constant terms \(4\) and \(-7\): \(4 - 7 = -3\). Thus, the expression simplifies to \(7x - 3\).

Key Concepts

Combining Like Terms

When simplifying expressions, one of the foundational skills is combining like terms. Like terms are terms within an expression that have identical variable parts, raised to the same power. This means that we can only combine terms that have the same variables, such as combining all terms with \(x\), all constants, and so forth.
For instance, in the expression \(7x - 3 + 4x\), the terms \(7x\) and \(4x\) are like terms because they both have the variable \(x\). The rule for combining these terms is straightforward: you add or subtract the coefficients (the numbers in front of the variables) and keep the common variable part. In this example, you would combine \(7x + 4x\) to get \(11x\).
Combining like terms is essential because it helps to simplify expressions, making them easier to work with or solve in the context of equations and inequalities. Whenever you simplify an expression, always be on the lookout for like terms to combine.

Expression Simplification

Expression simplification is the process of reducing an expression to its simplest form. This typically involves applying mathematical properties such as the distributive property, combining like terms as discussed, and performing basic arithmetic operations.
The main goal of simplification is to make the expression easier to understand and use in further calculations. In the given exercise, we started with \(4 + 7(x-1)\). Using the distributive property, we expanded \(7(x-1)\) to get \(7x - 7\). By substituting back and combining the constant terms \(4\) and \(-7\), we arrived at a simplified version: \(7x - 3\).
Expression simplification is crucial not only in algebra but also in calculus and beyond, as it allows mathematicians and students alike to manipulate and interpret expressions more efficiently.

Real Numbers

Real numbers encompass all the numbers you can think of: positive, negative, whole numbers, fractions, and decimals. They are an essential part of mathematics, forming the basis of many concepts in both simple arithmetic and advanced mathematics.
In the context of the problem, each component of the expression \(4\), \(7(x-1)\), and their resulting simplified form \(7x-3\) falls within the realm of real numbers. This property of real numbers assures that the arithmetic operations used to simplify expressions, like combining and distributing, remain valid and consistent.
Understanding real numbers is key to grasping more complex mathematical ideas. They provide a comprehensive framework that allows for representation and manipulation of quantities, making them integral to problem-solving and mathematical reasoning.