Problem 50 - Fundamentals of Mathematics

Question

For problems \(47-56\), simplify each expression by combining like terms. $$ -9 k-8 h-k+6 h $$

Step-by-Step Solution

Answer

The expression simplifies to \(-10k - 2h\).

Step 1: Identify Like Terms

In the expression \(-9k - 8h - k + 6h\), like terms are those with the same variable part. Here, like terms for \(k\) are \(-9k\) and \(-k\). Like terms for \(h\) are \(-8h\) and \(6h\).

Step 2: Combine Like Terms with 'k'

Add the coefficients of the \(k\) terms: \(-9k - k = -10k\). This simplifies that portion of the expression.

Step 3: Combine Like Terms with 'h'

Add the coefficients of the \(h\) terms: \(-8h + 6h = -2h\). This simplifies that portion of the expression.

Step 4: Write the Simplified Expression

Combine the results from previous steps to get the final simplified expression: \(-10k - 2h\).

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing a complex expression into its simplest form. This often entails combining like terms, eliminating any unnecessary components, and making the expression as straightforward as possible. To start, recognize the expression's different parts. These might include terms with variables, coefficients, and constants. By simplifying these components:

• The expression becomes easier to work with in equations.
• It improves computational efficiency and understandability.
• It allows for a clearer path to solving or graphing related problems.

In our example expression, \(-9k - 8h - k + 6h\), each term includes a variable part, either \(k\) or \(h\). Simplifying means we consolidate like terms while preserving the expression's mathematical integrity. Taking these steps helps in better understanding and handling algebraic problems.

Identifying Like Terms

Identifying like terms is a critical skill in algebra, as it sets the stage for simplifying expressions. Like terms are terms that have identical variable parts raised to the same power. In other words, they must share the same variable(s) and exponent(s) to be considered like terms.

• Examples of like terms are \(3x\) and \(-5x\), or \(2y^2\) and \(7y^2\).
• Non-examples include \(4x\) and \(4y\), or \(3x^2\) and \(5x\).

In our expression \(-9k -8h -k + 6h\), we can see that the terms \(-9k\) and \(-k\) are like terms because they both contain the variable \(k\). Similarly, \(-8h\) and \(6h\) are like terms as they both contain \(h\). Recognizing these pairs is crucial for combining them accurately. By focusing on the variable parts, you can smoothly navigate through complex algebraic expressions and solve them more efficiently.

Coefficients in Algebra

Coefficients are the numerical part of terms that multiply the variable(s). Understanding coefficients in algebra helps in manipulating and combining terms accurately. They tell us how many times a variable is accounted for in terms of its quantity or factor.

• In the term \(-9k\), \(-9\) is the coefficient of \(k\).
• In the term \(-8h\), \(-8\) is the coefficient.

Co-efficients might be positive or negative, and this directly affects the resultant operation when combining terms. In our sample expression,

• Addition occurs between coefficients of like terms when they are both positive or both negative.
• Subtraction occurs when they have opposite signs.

By knowing the role of coefficients, you can effectively combine like terms, as demonstrated when we combined \(-9k\) and \(-k\) to get \(-10k\), and \(-8h\) and \(6h\) to arrive at \(-2h\). Recognizing how coefficients influence the terms' sums and differences is a key part of simplifying algebraic expressions.