Problem 53 - Elementary Algebra

Question

For the following problems, use the distributive property to expand the quantities. $$b(r+5)$$

Step-by-Step Solution

Answer

Question: Use the distributive property to expand the given expression: \(b(r+5)\). Answer: The expanded expression is \(br+5b\).

Step 1: Identify the distributive property

We will use the distributive property, \(a(b+c) = ab + ac\), to expand the given expression \(b(r+5)\).

Step 2: Apply the distributive property

To expand \(b(r+5)\), multiply \(b\) by both terms inside the parentheses: $$b(r+5) = b \cdot r + b \cdot 5$$

Step 3: Simplify the expression

Now simplify each term in the expression: $$b \cdot r + b \cdot 5 = br + 5b$$ The expanded expression is: $$br+5b$$.

Key Concepts

Algebra

Algebra is a vast field of mathematics that focuses on using symbols and letters to represent numbers and quantities in formulas and equations. This branch of mathematics is fundamental to many real-world problems and advanced fields of study, and it involves various operations including addition, subtraction, multiplication, division, as well as more complex concepts such as factoring, exponents, and roots.

At its core, algebra teaches us how to solve for unknown values, known as variables. These variables are represented by letters, such as the 'b' and 'r' in the exercise example. Algebraic expressions often need to be simplified or expanded to solve equations or understand their components. The process of learning algebra involves various properties and rules that make it easier to work with these expressions, such as the distributive property which is crucial in expanding expressions.

Expanding Expressions

Expanding expressions is a fundamental technique in algebra where we rewrite a compacted equation or expression to its more detailed form, often by applying the distributive property. The distributive property states that for any real numbers a, b, and c, the expression 'a times (b plus c)' is equivalent to 'ab plus ac'.

To apply this property, we distribute the multiplication across the terms inside parentheses. For example, in our exercise, we expanded the expression by multiplying 'b' with both 'r' and '5' to obtain 'br plus 5b'. Expanding expressions is not just a procedural task but a concept that can reveal more information about the relationship between the variables involved and can simplify further calculations or the solving of equations.

Simplifying Expressions

Simplifying expressions is an essential algebraic skill that allows us to make expressions more manageable and easier to understand. It involves combining like terms, reducing fractions, and eliminating unnecessary parentheses. For simplifying, we follow certain mathematical rules and properties to ensure that the expression remains equivalent to its original form.

Once we've expanded an expression, like 'br plus 5b' from the exercise, the next step could be to simplify it if there are like terms to be combined. In our case, there are no like terms since 'br' and '5b' are not identical due to the presence of 'r'. However, simplification often follows expansion in more complex algebraic expressions, making this process a key tool in solving equations, proving identities, and performing operations with algebraic terms.