Problem 62 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 1(3 x+15)+2 x-12 $$

Step-by-Step Solution

Answer

Question: Simplify the following expression: (3x + 15) + 2x - 12 Answer: The simplified expression is 5x + 3.

Step 1: Distribute

Distribute the 1 to each term inside the parentheses: $$ (1 \cdot 3x) + (1 \cdot 15) + 2x - 12 $$

Step 2: Perform multiplication

Perform the multiplication operations: $$ 3x + 15 + 2x -12 $$

Step 3: Combine like terms

Combine the terms with the same variable (the x terms) and the constant terms (the numbers without variables): $$ (3x + 2x) + (15 - 12) $$

Step 4: Perform addition and subtraction

Perform the addition and subtraction operations: $$ 5x + 3 $$ The simplified algebraic expression is \(5x + 3\).

Key Concepts

Distributive Property

Understanding the distributive property is key to simplifying algebraic expressions. This property allows you to multiply a single term by each term within a parenthesis. For example, in the expression \(1(3x + 15)\), we distribute the multiplication of 1 across the terms inside the parenthesis, transforming it into \(3x + 15\). This may seem trivial with the number 1, but with any other number, it’s vital. For instance, if you had \(2(3x + 4)\), using the distributive property would give you \(6x + 8\).
In many algebra problems, you'll frequently distribute coefficients over terms to simplify expressions before combining like terms. It's essentially about ensuring each term is 'touched' by the multiplication, keeping the equation balanced and setting the stage for further simplification.

Combining Like Terms

Once we have distributed any coefficients, our next step often involves combining like terms. Like terms are terms that have the same variable raised to the same power. For instance, \(3x\) and \(2x\) are like terms because they both have the variable x raised to the first power. In the example \(3x + 15 + 2x - 12\), we can combine the like terms to further simplify the expression.
Here's how to do it:

• First, group the like terms: \((3x + 2x)\) and \((15 - 12)\).
• Second, simplify each group: \(3x + 2x\) becomes \(5x\), and \(15 - 12\) becomes 3.
• Last, put the simplified terms together to get the final expression: \(5x + 3\).

Understanding this concept is crucial for dealing with more complex algebraic equations, where simplifying expressions is a common step.

Algebraic Operations

Performing algebraic operations such as addition, subtraction, multiplication, and division with algebraic expressions is fundamental to the mastery of elementary algebra. After combining like terms, we often carry out these operations to simplify the expression further.
In the given problem, after distribution and combining like terms, we arrive at simple arithmetic: \(5x + 3\). This is the simplest form of the expression and cannot be reduced any further because it involves a variable term and a constant. It's important to follow the rules of arithmetic operations and to always perform operations on like terms to avoid mistakes in simplification.

Elementary Algebra

Elementary algebra is the branch of mathematics that deals with solving for the unknowns. It’s the bedrock upon which higher-level math builds. Simplifying algebraic expressions, like the one in our example, is a fundamental skill in algebra. It involves the use of distributive property, combining like terms, and performing algebraic operations, among other techniques.
To succeed in elementary algebra, it's essential to understand these foundational concepts and to practice applying them in various situations. This ensures a deep comprehension of how algebraic expressions are manipulated and sets the groundwork for tackling more complex problems in advanced math subjects.