Problem 2 - Intermediate Algebra

Question

Fill in the blanks. \(2 x+1=4\) is an example of a _____ equation in one variable.

Step-by-Step Solution

Answer

Linear

Step 1: Identify the form of the equation

The given equation is \( 2x + 1 = 4 \). This is written in the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants. This is typical for a certain type of algebraic equation.

Step 2: Recognize the characteristics

The equation has only one variable, \( x \), and the variable raised to the first power is indicative of a particular type of equation. Equations of this type involve a variable that can be solved straightforwardly without involving powers higher than one.

Step 3: Classify the type of equation

Equations in the form \( ax + b = c \) with one variable are known as linear equations. They graph as straight lines on a coordinate plane and are distinct from quadratic or other polynomial equations.

Key Concepts

One Variable Equations

When we talk about one-variable equations, we're referring to mathematical expressions that have exactly one unknown element which we usually represent using a letter like \( x \). In these equations, the variable appears only once and plays a crucial role in solving the equation.
The primary goal when working with one-variable equations is to determine the value of this variable that makes the equation true. To achieve this, you will manipulate the equation through algebraic methods to isolate the variable on one side of the equation.
For example, in the equation \( 2x + 1 = 4 \), the variable is \( x \). Since there's only one variable present, it's straightforward to solve using inverse operations. This simplicity makes one-variable equations an excellent starting point for learning basic algebra.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations like addition and subtraction. They form the building blocks of equations and inequalities. When you look at a one-variable equation like \( 2x + 1 = 4 \), you can see that this equation is made up of two expressions: \( 2x + 1 \) and \( 4 \).
A few key characteristics of algebraic expressions include:

• Variables: These are symbols that represent unknown values. In our example, \( x \) is the variable.
• Constants: These are fixed numbers. Here, \( 1 \) and \( 4 \) are constants.
• Coefficient: This is a number that multiplies the variable, such as the \( 2 \) in \( 2x \).

Understanding these components helps in simplifying and solving algebraic equations.

Solving Equations

Solving equations, especially linear equations in one variable \( ax + b = c \), involves finding the value of the variable that makes the equation true. This process often includes the following steps:

• Isolate the variable: Use inverse operations to get the variable by itself on one side of the equation. For example, in \( 2x + 1 = 4 \), we subtract \( 1 \) from both sides to get \( 2x = 3 \).
• Eliminate the coefficient: Divide both sides by the coefficient of the variable to solve for the variable. In this case, dividing by \( 2 \) gives \( x = \frac{3}{2} \).
• Verification: Substitute the found value back into the original equation to ensure it satisfies the equation.

These steps will help you solve not just one-variable linear equations, but also set a foundation for more complex algebraic manipulations later on.