Problem 17 - Fundamentals of Mathematics

Question

Simplify each expression by combining like terms. $$15 r-6 s+2 r+8 s-6 r-7 s-s-2 r$$

Step-by-Step Solution

Answer

The simplified expression is \( 9r - 6s \).

Step 1: Identify Like Terms

To simplify the expression, first identify the like terms. Like terms have the same variable raised to the same power. In the expression, the terms with 'r' are \( 15r, 2r, -6r, -2r \) and the terms with 's' are \( -6s, 8s, -7s, -s \).

Step 2: Combine 'r' Terms

Add the coefficients of the 'r' terms together. This gives us: \( 15r + 2r - 6r - 2r = (15 + 2 - 6 - 2)r = 9r \).

Step 3: Combine 's' Terms

Next, add the coefficients of the 's' terms: \( -6s + 8s - 7s - s = (-6 + 8 - 7 - 1)s = -6s \).

Step 4: Write the Simplified Expression

Combine the results from steps 2 and 3 to get the simplified expression: \( 9r - 6s \).

Key Concepts

Combining Like Terms

Combining like terms is a fundamental skill in simplifying algebraic expressions. It involves adding or subtracting terms that have identical variables and exponents. Think of like terms as similar items that can be grouped together.

• For example, in the expression \( 3x + 4x \), both terms have the variable 'x', making them like terms. They can be added together to form \( 7x \).
• This process helps streamline expressions, making them easier to read and solve. The goal is to condense the expression into its simplest form.

When combining terms, only work with the coefficients (the numerical part before the variable) while keeping the variable and exponent unchanged. Remember that terms like \( 3y \) and \( 2y^2 \) are not like terms because they have different exponents.By practicing this process, the solving of algebra problems becomes more intuitive and less cumbersome.

Identifying Like Terms

Identifying like terms comes before you can successfully combine them. The key is spotting variables that are the same, along with any exponents.

• In the exercise \(15 r - 6 s + 2 r + 8 s - 6 r - 7 s - s - 2 r\), the task is to pinpoint which terms belong together.
• The terms with the variable 'r', including \(15r, 2r, -6r,\) and \(-2r\), can be grouped because they share the same variable.
• Likewise, terms with 's', such as \(-6s, 8s, -7s,\) and \(-s\), make another group suitable for combining.

This step is vital as errors in grouping can lead to incorrect simplification, affecting overall problem-solving accuracy. It's all about attention to details—watch those variables and their exponents closely!

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They serve as the foundation for algebra and are used to represent real-world situations in a mathematical form.

• An expression like \( 15r - 6s + 2r + 8s - 6r - 7s - s - 2r \) combines constants and variables, with operations dictating their relationships.
• Understanding the structure of algebraic expressions can help in manipulating and simplifying them.

When working with these expressions, the absence of an equals sign means they are not equations and are open to simplification but not solving. Grasping the concept of terms (like \(15r\) or \(-6s\) ) within an expression is important to tackle more complicated algebra problems later on. Simple expressions can build up to complex equations, requiring a solid understanding of foundational concepts.