Problem 5 - Algebra 1: Concepts and Skills

Question

Identify the coefficient of each variable term. $$ 4 x-2 x=6 $$

Step-by-Step Solution

Answer

The coefficients of the variable 'x' in the equation \(4x - 2x = 6\) are 4 and -2 respectively.

Step 1: Identify the variable terms

In this equation, there are two variable terms: \(4x\) and \(-2x\). Both terms include the variable 'x' and are thus identified as variable terms.

Step 2: Identify coefficients of the variable terms

The coefficient of the variable term is the number that multiplies the variable. In \(4x\), the coefficient of 'x' is 4. Similarly, in \(-2x\), the coefficient of 'x' is -2. So the coefficients we've found for the variable 'x' in this equation are 4 and -2.

Key Concepts

Variable Terms

In order to understand algebra, it's crucial to become familiar with variable terms. A variable term is composed of a number, known as a coefficient, and a variable, which is typically represented by a letter that stands for an unknown value. An example of this would be the term \(4x\), where 4 is the coefficient and \('x'\) is the variable.

In the given exercise, \(4x\) and \(\minus2x\) are both variable terms because they include a variable part (\('x'\)) and a numerical part. When handling algebraic expressions, recognizing these variable terms is essential since they are the building blocks of algebra equations. To simplify or solve equations, it's often necessary to combine like terms, which are variable terms with the same variable raised to the same power.

Identifying Coefficients

Identifying coefficients within algebraic expressions allows you to understand the relationship between variables and their numerical multipliers. As seen in the exercise, a coefficient is the number directly multiplied by the variable within a term.

For instance, if we examine the term \(4x\), we observe that the number 4 is the coefficient of the variable \(x\). Coefficients can be positive or negative, whole numbers, fractions, or decimals. It's imperative to pay attention to the sign of the coefficient as well. In the term \(\minus2x\), the coefficient is \(\minus2\),not just 2, indicating it is negative. Therefore, in the step-by-step solution, the first step we take is to find the coefficients of all variable terms in the equation. Being accurate with coefficients is critical because they determine the term's value and affect the outcome when solving equations.

Algebraic Expressions

Algebraic expressions are the centerpiece of algebra and consist of numbers, variables, and operation symbols such as plus, minus, times, and divided by. They do not contain an equals sign, as equations do. An expression can be simple, consisting of a single term like \(5y\), or more complex with several terms such as \(4x - 2x \+ 7\).

In this exercise, \(4x - 2x\) is an algebraic expression that is made up of two variable terms. Algebraic expressions can be manipulated by combining like terms, expanding using the distributive property, and factoring, among other operations. The goal is often to simplify the expression to reveal more information about the variables or to make it easier to solve an associated equation. With practice, maneuvering within the nuances of these expressions can become intuitive, leading to a better understanding of handling algebraic equations.