Problem 18 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate each expression for \(x=7\) and \(y=5\). $$3(x+y)$$

Step-by-Step Solution

Answer

So the evaluated expression \(3(x+y)\) for \(x=7\) and \(y=5\) is \(36\).

Step 1: Substitute given values

Replace \(x\) and \(y\) in the expression \(3(x+y)\) with the given values \(7\) and \(5\) respectively. Therefore, it becomes \(3(7+5)\).

Step 2: Perform arithmetic operations

Simplify the parentheses first according to the BIDMAS/BODMAS rule, which guides the order of operations. i.e. Brackets/Order (powers and square roots, etc.), Division and Multiplication (left-to-right), Addition and Subtraction (left-to-right). So \(7+5\) becomes \(12\). Therefore, the expression now turns into \(3 \times 12\). Multiply \(3\) by \(12\) to get \(36\).

Key Concepts

Order of Operations

Understanding the order of operations is crucial in algebra as it ensures that everyone interprets mathematical expressions consistently. The order of operations, often remembered using the acronym BIDMAS or BODMAS, dictates the sequence in which calculations should be carried out:

• Brackets first: Solve expressions within parentheses or brackets.
• Orders (also known as exponents or powers): Evaluate powers and roots.
• Division and Multiplication: Perform these operations next, moving from left to right.
• Addition and Subtraction: Finally, carry out addition and subtraction, moving left to right as well.

When evaluating the expression \(3(x+y)\), we first need to calculate within the brackets. Thus, add \(x\) and \(y\) to follow the correct order. In our case, after substituting the values of \(x\) and \(y\), we first perform the operation inside the brackets (\(7+5\)) before proceeding with multiplication by 3. This method ensures accurate results.

Substitution in Expressions

Substitution is a basic yet essential concept in algebra, where you replace variables with given or known values in an expression. This step involves taking the provided values for variables and plugging them into the formula or expression, which simplifies the expression into a manageable arithmetic problem.

In the example provided, we receive the values \(x=7\) and \(y=5\) to substitute into the expression \(3(x+y)\). The substitution step is crucial because it transforms an abstract expression into something tangible and solvable. After substituting, the expression \(3(x+y)\) becomes \(3(7+5)\), making it significantly simpler to calculate by eliminating the variables.

Arithmetic Operations

Arithmetic operations form the foundation of calculating expressions in mathematics. They are the basic operations, including addition, subtraction, multiplication, and division, that you likely learned early on in your math studies. These operations are tools for moving from a simplified expression to an actual number.

In the given example, once we've completed substitution and followed the order of operations, we perform a series of arithmetic steps. Initially, after substituting \(x\) and \(y\), the expression \(3(7+5)\) involves adding 7 and 5, which equals 12. Following this, we multiply 3 by the resultant sum, which is the arithmetic operation necessary to conclude the problem, giving us the final result of 36. Each step builds incrementally, using basic arithmetic to arrive at the solution.