Problem 114 - Algebra and Trigonometry Real Mathematics, Real People

Question

Identify the terms. Then identify the coefficients of the variable terms of the expression. $$10+3 x$$

Step-by-Step Solution

Answer

The terms of the expression are 10 and \(3x\). The coefficient of the variable term \(x\) is 3.

Step 1: Identify the terms

The expression is \(10+3x\). Here, the terms are separated by the plus sign. Thus, the terms of the expression are 10 and \(3x\).

Step 2: Identify the coefficients of the variable terms

A coefficient in a term is the number that is multiplied by a variable in the term. The variable term here is \(3x\). Therefore, the coefficient of the variable term \(x\) is 3.

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. Variables represent unknown values and are often denoted by letters, like x, y, or z. Operation symbols include addition (+), subtraction (-), multiplication (*), and division (/). In the expression \(10 + 3x\), we have a combination of numbers and a variable along with an addition sign.

Understanding algebraic expressions is crucial because they are the foundational building blocks for forming equations and inequalities, which can be used to solve real-world problems. In classroom settings, learning to recognize and manipulate these expressions helps students develop their mathematical thinking and prepares them for more complex concepts in algebra.

Coefficients

In algebra, coefficients are the numerical part of the terms that are multiplied by the variables in an algebraic expression. They provide us with critical information on how the variable affects the overall value of the expression. For instance, if we examine the term \(3x\) from the expression \(10 + 3x\), the number 3 is the coefficient of the variable x. This indicates that x is being multiplied by 3.

Coefficients can be positive or negative and they determine the rate at which the value of the expression changes as the variable changes. They are essentially the 'scaling factor' that affects the magnitude of the variable term. Understanding coefficients is also valuable when comparing the relative influence of different variable terms within the same expression or across different expressions.

Variable Terms

Variable terms are parts of algebraic expressions that contain both a variable and, usually, a coefficient. In the given expression \(10+3x\), the variable term is \(3x\). This term is comprised of the variable x and its coefficient 3. Variable terms are what make expressions dynamic because the value of the term changes with the value of the variable.

Identifying variable terms in an expression is a foundational skill in algebra. It enables students to simplify expressions, solve equations, and understand how different parts of an expression contribute to its overall value. Recognizing and working with variable terms helps in discovering patterns and in creating equations that model real-world scenarios.