Problem 20 - Fundamentals of Mathematics

Question

Simplify each expression by combining like terms. $$(-4+1) k+(6-3) k+(12-4) h+(5+2) k$$

Step-by-Step Solution

Answer

The simplified expression is \(7k + 8h\).

Step 1: Simplify the Constants Inside the Parentheses

First, simplify each arithmetic operation inside the parentheses. For \[(-4+1)k = (-3)k\] \[((6-3)k = 3k)\] \[((12-4)h = 8h)\] and \[((5+2)k = 7k)\]. So, the expression becomes \[-3k + 3k + 8h + 7k\].

Step 2: Combine Like Terms Involving 'k'

Now, combine the like terms that have 'k' as a factor. In the expression, we have \[-3k, 3k, \text{and } 7k\]. Combine them: \[-3k + 3k + 7k = 7k\].

Step 3: Combine All Terms

Finally, write down the simplified expression with the combined terms. After combining the 'k' terms, the simplified expression is:\[7k + 8h\].

Key Concepts

Simplification

Simplification is all about making expressions easier to understand and use. Think of it as a way to clean up messy math problems. When we simplify an algebraic expression, we are aiming to write it in the smallest, easiest form possible. It involves removing any unnecessary elements and combining like parts together.
To simplify an expression, always start by performing basic operations within any parentheses, much like in our example where we started with \((-4 + 1)k, (6 - 3)k, (12 - 4)h, ext{and } (5 + 2)k\). This initial step helps to minimize complexity early on.

• First, resolve operations within parentheses.
• Next, combine like terms (terms with the same variables and exponents).

Simplification doesn’t change the value of the expression but rephrases it in a friendlier form, which is a powerful tool when solving equations or evaluating functions.

Algebraic Expressions

Algebraic expressions represent numbers and operations in a flexible form, using variables and constants. In our example, terms like \(-3k, 3k, ext{and } 8h\) are all algebraic expressions. Here, 'k' and 'h' are variables representing unknown numbers, whereas numerical coefficients like -3, 3, and 8 are constants.
These expressions do not have an equal sign, unlike equations. They can be manipulated using arithmetic operations to simplify or evaluate under particular conditions.
An important aspect of working with algebraic expressions is recognizing like terms, which are terms that have the exact same variable raised to the same power. For example, \(-3k, 3k, ext{and } 7k\) are like terms, allowing us to combine them for simplification.

• Use algebraic expressions to model real-world situations mathematically.
• Combine only like terms to simplify an expression.

Arithmetic Operations

Arithmetic operations include basic math processes such as addition, subtraction, multiplication, and division. In algebra, these operations are used not just with numbers but with variables and expressions as well.
In the given example, arithmetic operations were used to simplify values within parentheses first, such as calculating \(-4 + 1\).
When working with arithmetic operations in algebra:

• Always resolve operations within parentheses first, according to the order of operations.
• Add and subtract like terms to simplify the expression.

Remembering the hierarchy of operations, often abbreviated as PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction), can guide you in correctly executing these operations on both numerical and algebraic expressions.