Problem 14 - Fundamentals of Mathematics

Question

Simplify each expression by combining like terms. $$7 h+3 a-10 k+6 a-2 h-5 k-3 k$$

Step-by-Step Solution

Answer

The simplified expression is \(5h + 9a - 18k\).

Step 1: Identify Like Terms

The given expression is \(7h + 3a - 10k + 6a - 2h - 5k - 3k\). Start by identifying the like terms. Like terms are terms that have the same variable raised to the same power. In this case, group terms involving \(h\), \(a\), and \(k\).

Step 2: Combine Like Terms for h

For terms involving \(h\), combine \(7h\) and \(-2h\). Calculate: \(7h - 2h = 5h\). The combined term for \(h\) is \(5h\).

Step 3: Combine Like Terms for a

For terms involving \(a\), combine \(3a\) and \(6a\). Calculate: \(3a + 6a = 9a\). The combined term for \(a\) is \(9a\).

Step 4: Combine Like Terms for k

For terms involving \(k\), combine \(-10k\), \(-5k\), and \(-3k\). Calculate: \(-10k - 5k - 3k = -18k\). The combined term for \(k\) is \(-18k\).

Step 5: Write the Simplified Expression

Combine the simplified terms for \(h\), \(a\), and \(k\) to write the final answer: \(5h + 9a - 18k\).

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential skill in algebra. It makes complex expressions easy to understand and work with. Here's how it works: when you simplify an algebraic expression, you condense it to the simplest form without changing its value. This involves removing any unnecessary terms and operations, such as performing addition or subtraction between like terms.

• Firstly, study the expression carefully to identify any terms that can be simplified.
• Next, look for operations that can be performed, like adding and subtracting similar terms.
• Finally, write down the expression in its simplest form.

Remember: simplification doesn't alter the intrinsic value, just how it looks.

Identifying Like Terms

Identifying like terms is a key step in manipulating algebraic expressions. Like terms are those that have the same variable part, which means they have similar variables raised to the same power.

• Even if coefficients (the numbers in front of the variables) differ, as long as the variables match, the terms are considered like terms.
• For instance, in the expression \(7h, -2h,\) and \(3a, 6a\), terms involving the same letter are like terms.
• Once like terms are identified, they can be combined by adding or subtracting the coefficients.

Being able to identify and combine like terms simplifies expressions effectively, which is central to many algebra problems.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. Understanding these is crucial, as they form the backbone of algebra. Let's take a closer look:

• They may vary in complexity from simple to compound expressions, involving numerous terms and operations.
• The key components include constants (numbers without variables), variables (letters that represent numbers), coefficients (numbers multiplied by variables), and operators (like +, -, *, and /).
• Understanding how to manipulate algebraic expressions forms the basis for more advanced topics in algebra.

Learning to recognize and simplify these expressions allows you to solve equations more smoothly.