Problem 7 - College Algebra

Question

For the following exercises, solve the equation for \(x\). $$ 4 x-3=5 $$

Step-by-Step Solution

Answer

The solution is \(x = 2\).

Step 1: Understand the Equation

The given equation is \(4x - 3 = 5\). Our goal is to solve for \(x\), which means finding the value of \(x\) that makes the equation true.

Step 2: Isolate the Variable

To solve for \(x\), we need to isolate it on one side of the equation. Start by adding 3 to both sides to cancel out the -3. This gives us: \(4x - 3 + 3 = 5 + 3\). Simplifying this, we have \(4x = 8\).

Step 3: Solve for x

Now, divide each side of the equation by 4 to solve for \(x\). This results in: \(\frac{4x}{4} = \frac{8}{4}\). Simplifying this, we find \(x = 2\).

Key Concepts

Solving Equations

When tackling an algebraic equation, the main aim is to find the value of the unknown variable, such as \(x\), that satisfies the equation. Solving equations requires applying mathematical operations in a logical sequence.

This process generally involves:

• Understanding the structure of the equation.
• Applying operations to isolate the variable.
• Performing arithmetic to solve for the variable.

For example, in the equation \(4x - 3 = 5\), we begin by reading and comprehending that \(x\) is what we need to find. The logic flows from transforming the equation step-by-step until the answer is revealed. Each move keeps the balance of the equation intact, ensuring both sides remain equal.

Remember, whatever operation is performed on one side must also be done to the other side — this is the golden rule for solving equations. It's not only about the operations themselves but about maintaining balance throughout the process.

Isolate the Variable

Isolating the variable is a crucial step in solving the equation. The aim is to have the variable you're solving for, often \(x\), on one side of the equation by itself.

Here's how you do it:

• Start by looking at the equation and determine which terms should be moved to the other side to get \(x\) by itself.
• In our example, \(4x - 3 = 5\), we first need to eliminate the \(-3\).
• This is done by adding 3 to both sides of the equation: \(4x - 3 + 3 = 5 + 3\).

Once simplification is complete, we end up with \(4x = 8\), leaving \(x\) ready to be completely isolated. By doing this, you prepare the stage to directly calculate the value for \(x\). Each step to isolate the variable should be simple, logical, and run smoothly.

Linear Equations

Linear equations are ones that form a straight line when graphed. They usually take the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants. Solving linear equations involves straightforward operations that linearly affect the variable.

In our specific case, \(4x - 3 = 5\) is a linear equation. The term "linear" means you are working with an equation that relates to simple, direct relationships between variables and constants.

Characteristics of linear equations include:

• There are no exponents, meaning the variable is not raised to any power higher than one.
• Each term is either a constant or the product of a constant and a single variable.
• Steps to solve these involve basic arithmetic and logical operations, typically resulting in a straightforward solution.

Linear equations like this provide a foundational understanding of more complex algebraic concepts yet are essential to mastering for anyone looking to advance in mathematics.