Problem 17 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate each expression for \(x=7\) and \(y=5\). $$2(x+y)$$

Step-by-Step Solution

Answer

The evaluated expression is 24.

Step 1: Substitute the given values

Replace \(x\) with 7 and \(y\) with 5 in the given expression. This will turn \(2(x+y)\) into \(2(7+5)\).

Step 2: Apply the order of operations

According to the order of operations in mathematics, perform the operation in the parentheses first. This changes \(2(7+5)\) into \(2 * 12\).

Step 3: Perform the multiplication

Perform the multiplication step to arrive at the final answer. This changes \(2 * 12\) into 24.

Key Concepts

Understanding the Order of Operations

As you delve into the world of algebra, grasping the order of operations is crucial. It's the set of rules that mathematicians agree upon to perform calculations consistently. Think of it like the rules of the road but for math—without them, things would be chaotic. When evaluating any mathematical expression, begin by doing the operations enclosed in parentheses or brackets. Next, tackle exponents or powers. After that, perform multiplication and division from left to right, and finally, address addition and subtraction from left to right. This sequence can be remembered with the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Applying the order of operations to our exercise, the first step is to calculate the sum within the parentheses (7+5), before multiplying it by 2. Getting this right ensures that expressions are evaluated accurately, every single time.

Employing the Substitution Method

The substitution method is essentially a way of replacing variables with their known values. By substituting, we can simplify an expression or an equation and make it solvable. Imagine variables as placeholders for numbers—they can represent any value. When the value is known, like in our exercise with the values of 'x' being 7 and 'y' being 5, we swap the variables for these real numbers.

In practical terms, here's how we applied the substitution method to the expression: replace 'x' with 7 and 'y' with 5 in the expression to transform it. This simple replacement turns the abstract into something concrete, thus making it possible to work out the exact value of the expression.

Exploring Algebraic Expressions

Algebraic expressions form the foundation of algebra. They are made up of variables, numbers, and arithmetic operations. Consider them as phrases in the language of mathematics that help describe patterns or general relationships between quantities. These expressions might include single variables (like 'x' or 'y'), constants (specific numbers), and various operations (such as addition, subtraction, multiplication, and division).

An algebraic expression doesn't have an equals sign, unlike an equation; hence it doesn't present a concrete solution but instead represents a value based on the input provided. For example, in the expression from our exercise, '2(x+y)', the 'x' and 'y' are variables that can change, while the number 2 and the operations of addition and multiplication are fixed. Learning to work with algebraic expressions paves the way to understanding more complex mathematical concepts.