Problem 79 - Algebra and Trigonometry Real Mathematics, Real People

Question

Identify the rule of algebra illustrated by the statement. \(5(a-9)=5 a-45\)

Step-by-Step Solution

Answer

The rule of algebra illustrated by the statement \(5(a-9)=5 a-45\) is the Distributive Property of Multiplication over Subtraction.

Step 1: Observe the original equation and divide the expression

The original equation is \(5(a-9)=5 a-45\). On the LHS, it's apparent that the term \(5\) is being multiplied with the quantity \((a-9)\). On the RHS, we see two separate terms, \(5 a\) and \(-45\). Begin by dividing the expression on the LHS.

Step 2: Compare both sides of the equation

After dividing the term on the left side, we get \(5a - 45\). Now, both sides of the equation match, i.e., \(5a - 45 = 5a -45\). The exact same transformation has been done on both sides of the equation

Step 3: Identify the rule of algebra

Comparing the original equation and the simplified equation, it's clear that the term \(5\) has been multiplied by each term inside the parentheses \((a-9)\). This operation of first multiplying \(5\) by \(a\) and then by \(-9\) is actually demonstrating the 'Distributive Property of Multiplication over Subtraction'.

Key Concepts

Algebraic Expressions

Algebraic expressions are the cornerstone of algebra and are comprised of variables, numbers, and operations. In an expression like \(5(a-9)\), we have a variable \(a\), a number \(5\), and an operation \(\) (multiplication over subtraction in this case). Such expressions can often be simplified by applying different algebraic rules, which can make them easier to understand or manipulate for further calculations.

When we talk about simplifying algebraic expressions, we often look for ways to make them as 'neat' as possible—combining like terms, eliminating unnecessary brackets, and factoring, are some of the go-to strategies in this process. Understanding how to manipulate these expressions correctly is essential for solving equations, evaluating functions, and graphing linear relationships among many other mathematical applications.

Equation Simplification

Simplifying an equation involves reducing it to its simplest form while keeping it equivalent to the original. Similar to tidying a cluttered room, equation simplification makes it easier to see what exactly you're working with and lays the groundwork for solving it efficiently. In our example \(5(a-9)=5a-45\), simplification doesn't change the equality; It merely rearranges it in a more digestible manner.

The equation simplification often starts with the distributive property and can also include combining like terms and reducing fractions to their lowest terms. Simplifying equations is a critical skill because it not only helps in solving problems but also in understanding the fundamental properties and relationships between mathematical expressions.

Multiplication over Subtraction

Multiplication over subtraction is an application of the distributive property, which states that multiplying a number by a difference is the same as multiplying the number by each term and then subtracting the results. Mathematically, this is represented as \(a(b - c) = ab - ac\).

In our exercise, \(5(a-9)\) demonstrates this property. Here’s how it works using the steps noted in the solution:

Apply the Distributive Property

To distribute \(5\) over \(a-9\), multiply \(5\) times each term inside the parentheses: \(5 \times a\) and \(5 \times -9\).

Simplify the Result

We then simplify to get \(5a - 45\), which matches the right-hand side (RHS) of the equation. This shows that \(5(a - 9)\) is indeed equal to \(5a - 45\), verifying our understanding of the distributive property in action.