Problem 8 - Algebra 1: Concepts and Skills

Question

Identify the coefficient of each variable term. $$ 2(x+1)=14 $$

Step-by-Step Solution

Answer

The coefficient of the variable term in the equation \(2(x+1)=14\) is 2.

Step 1: Expand the Expression

Let's start by multiplying the equation through the brackets. This will simplify the equation. \(2 \cdot x + 2 \cdot 1 = 14\).

Step 2: Simplify the Equation

After calculating the multiplication, the equation simplifies to \(2x + 2 = 14\).

Step 3: Identify the Coefficient

For the term \(2x\), the coefficient of the variable x is 2. Coefficient is the numerical factor of a term that contains a variable. Here, the coefficient is 2.

Key Concepts

Understanding Coefficient in Algebraic Expressions

When dealing with algebraic expressions, you often encounter terms that consist of numbers and variables. The coefficient is the numerical part of these terms. For example, in the term \(2x\), the number \(2\) is called the coefficient of \(x\). It represents how many times the variable \(x\) is multiplied.
A coefficient can be positive or negative, whole number or a fraction. If a term is written as just a variable, like \(y\), the coefficient is implied to be 1 since \(y = 1y\).
Recognizing coefficients is important because they tell you the relationship between the number and the variable. This understanding helps simplify expressions and solve equations efficiently.

Expanding Brackets in Equations

Expanding brackets is a key step in simplifying and solving equations. It involves distributing the number outside the bracket to each term inside the bracket. This process converts an expression like \(2(x + 1)\) into \(2x + 2\).
To perform this multiplication properly, follow these simple steps:

• Multiply the external number by the first term inside the bracket.
• Repeat this action for every term inside the bracket.
• Ensure you apply the appropriate sign to ensure addition or subtraction is handled correctly.

Expanding brackets simplifies the structure of an equation, allowing us to further manipulate it and isolate variables. It makes the equation easier to handle, setting the stage for further simplification or solving.

Simplifying Equations for Solving

Simplifying an equation means transforming it into its simplest form, making it easier to solve or analyze. After expanding brackets, as illustrated in the equation \(2x + 2 = 14\), the goal is often to isolate the variable.
To simplify, you can

• Combine like terms: Terms that have the same variables raised to the same power can be summed or subtracted.
• Eliminate constant terms from one side of the equation, usually achieved through addition or subtraction.
• Divide or multiply the whole equation to isolate the variable, particularly when coefficients are involved.

Through these steps, complex equations become straightforward, enabling you to find solutions effectively and accurately. Simplifying is an essential skill that underpins successful problem-solving in algebra.