Problem 22 - Intermediate Algebra

Question

Solve each equation. Check each result. See Example 2. $$ 3 x+1=3 $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{2}{3}\).

Step 1: Isolate the Variable on One Side

Start with the equation: \(3x + 1 = 3\). To isolate \(x\), you need to eliminate the constant term on the left side. So, subtract 1 from both sides of the equation to remove the constant:\[3x + 1 - 1 = 3 - 1\]This simplifies to:\[3x = 2\]

Step 2: Solve for x

Now, divide both sides of the equation by 3 to solve for \(x\):\[\frac{3x}{3} = \frac{2}{3}\]This results in:\[x = \frac{2}{3}\]

Step 3: Check Your Result

To verify the solution, substitute \(x = \frac{2}{3}\) back into the original equation \(3x + 1 = 3\):\[3\left(\frac{2}{3}\right) + 1 = 3\]This simplifies to:\[2 + 1 = 3\]Since both sides of the equation are equal, \(x = \frac{2}{3}\) is indeed the correct solution.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in mathematics. It involves finding the value of a variable that makes an equation true. In other words, we want to know "what number can replace the variable to balance both sides of the equation?" The process involves several steps aimed at simplifying the equation until the variable is isolated on one side. Let's take a closer look at how this works. Consider simplifying the equation:

• Start with the given equation. For example, we have: \(3x + 1 = 3\).
• Aim to get all terms involving the variable on one side and constants on the other.
• Use basic arithmetic operations like addition, subtraction, multiplication, and division to simplify.

In this process, we systematically manipulate the equation to simplify it while keeping it balanced. Each operation must be performed on both sides of the equation to maintain equality. The ultimate goal is to find a value for \(x\) that satisfies the equation.

Isolation of Variable

Isolation of a variable is about making one side of an equation purely the variable, while everything else is moved to the other side. This technique simplifies the process of finding the variable's value.To isolate a variable:

• Identify the term containing the variable you want to solve for. In our example, this is \(3x\).
• Remove any constants from the variable's side by reversing arithmetic operations. If a constant is added to the variable's term, subtract it from both sides. If it is subtracted, add it back. In the equation \(3x + 1 = 3\), subtract \(1\) from both sides to get \(3x = 2\).
• Next, remove any coefficients (numbers multiplying the variable) by performing the inverse operation. For \(3x = 2\), divide both sides by \(3\) to isolate \(x\), resulting in \(x = \frac{2}{3}\).

This process requires careful attention to detail, ensuring that each arithmetic operation is correctly applied to both sides of the equation. This maintains the balance and integrity of the equation throughout the solution.

Checking Solutions

Once you have a proposed solution, checking it is crucial to ensure its accuracy. This helps you verify that the steps you performed are correct and that the solution truly satisfies the original equation.To check solutions:

• Substitute the proposed solution back into the original equation. For example, we found \(x = \frac{2}{3}\).
• Replace the variable in the original equation with this value: \(3\left(\frac{2}{3}\right) + 1\).
• Simplify both sides of the equation: \(2 + 1 = 3\).
• Confirm whether both sides of the equation are equal. If they are, the solution is correct. \(2 + 1\) indeed equals \(3\), so \(x = \frac{2}{3}\) is correct.

Checking solutions is an essential step that provides confidence in the accuracy of your solution and ensures that no mistakes were made during the solving process.