Problem 86 - Algebra A Combined Function

Question

Write each algebraic expression described. Do like terms always contain the same numerical coefficients? Explain your answer.

Step-by-Step Solution

Answer

No, like terms only need the same variable parts, not the same coefficients.

Step 1: Understanding Like Terms

Like terms in algebra are terms that have the same variable raised to the same power. For example, in the expression \( 3x + 5x \), the terms \( 3x \) and \( 5x \) are like terms because they both contain the variable \( x \) raised to the same power (which is 1 in this case).

Step 2: Examining Coefficients

Numerical coefficients are the numerical part of terms that are multiplied by the variables. In the expression \( 3x + 5x \), the coefficients are 3 and 5, respectively. Coefficients can differ even between like terms because the definition of like terms only requires matching variable parts, not matching coefficients.

Step 3: Concluding Explanation

Like terms do not need to have the same numerical coefficients. The key requirement is that they have the same variable parts. The coefficients can vary, as seen in standard operations where we combine like terms by adding or subtracting their coefficients.

Key Concepts

Algebraic Expression

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. Unlike an equation, an algebraic expression does not have an equal sign. It is simply a way to represent a mathematical situation. For example, the expression \( 2x + 5 \) represents "two times a variable \( x \) plus five." You might come across algebraic expressions in various forms, such as linear, quadratic, or more complex ones. Each part of an algebraic expression, like the numbers and the variables, plays a specific role in solving problems and understanding mathematical relationships.

Numerical Coefficients

Numerical coefficients are the numbers in front of the variables in an algebraic expression. They serve as multipliers of the variables. For instance, in the expression \( 4y - 3z + 7 \), the numerical coefficients are 4 and -3. The number 7 is a constant term and does not have a variable multiplied by it. Understanding numerical coefficients is crucial because they help determine the value of an expression when the variables are assigned particular values. They give size or magnitude to the variable terms they accompany.

• A positive coefficient increases the term's value as the variable increases.
• A negative coefficient decreases the term's value as the variable increases.

Hence, knowing how coefficients influence the expression helps you solve and manipulate equations effectively.

Variable

In mathematics, a variable is symbolized, typically by a letter, and represents a number that can change or vary. A variable allows expressions and equations to be more flexible, letting us represent and solve a wide range of problems. In the algebraic expression \( 7a + 3b - 5 \), \( a \) and \( b \) are variables. They can represent any number. This property makes variables powerful tools in mathematics:

• They help in forming general statements and formulas.
• They allow for the expression of relationships between changing quantities.

Understanding variables and their roles in algebraic expressions is essential for studying how different mathematical components relate and interact.

Combining Like Terms

Combining like terms involves simplifying expressions by adding or subtracting terms whose variables and exponents are the same. This process is crucial in simplifying algebraic expressions and solving equations. For instance, in the expression \( 6x + 3 + 2x - 5 \), the like terms are \( 6x \) and \( 2x \), which can be combined to become \( 8x \). Similarly, the constants \( 3 \) and \( -5 \) can be combined, resulting in the simplified expression \( 8x - 2 \). To identify like terms:

• Check if the terms have the same variable raised to the same power.
• Don't worry about their coefficients at first, as these can be different.

By focusing on the variable part, you can easily recognize which terms can be combined to simplify your work in solving algebraic problems.