Problem 52 - Elementary Algebra

Question

The difference of \(3 x\) and 8 is \(25 .\)

Step-by-Step Solution

Answer

The value of \(x\) is 11.

Step 1: Translate the problem into an equation

The problem states that the difference between \(3x\) and 8 is 25. This can be expressed as an equation: \(3x - 8 = 25\).

Step 2: Isolate the variable term

To isolate the term \(3x\), add 8 to both sides of the equation to balance it. This gives: \[3x - 8 + 8 = 25 + 8\] Simplifying both sides, we get:\[3x = 33\]

Step 3: Solve for x

To find the value of \(x\), divide both sides of the equation by 3, which is the coefficient of \(x\). This gives:\[x = \frac{33}{3}\] Simplifying the division, we obtain:\[x = 11\]

Step 4: Verify the solution

Substitute \(x = 11\) back into the original equation to ensure it holds true:\[3(11) - 8 = 25\] Calculating the left side: \[33 - 8 = 25\] Since both sides of the equation are equal, the solution \(x = 11\) is correct.

Key Concepts

Solving Equations

When we talk about solving equations, we are referring to the process of finding the value of a variable that makes an equation true. Equations are mathematical statements where two expressions are set equal to one another. For example, the equation given in the exercise is \(3x - 8 = 25\). This means that we need to determine the value of \(x\) that satisfies this equality.
To solve equations effectively, it's important to understand what the equation represents and to systematically work through to the solution. Here’s a simple process to follow:

• Identify the equation from the problem statement.
• Determine what mathematical operations are involved.
• Apply inverse operations to simplify each side of the equation until the variable is isolated.

By focusing on these steps, you ensure the process is organized and straightforward.

Variable Isolation

Variable isolation is a crucial step in solving equations. It involves manipulating the equation in such a way as to get the variable by itself on one side of the equation. Let's consider our equation, \(3x - 8 = 25\).
The goal is to have \(x\) alone on one side. Here’s how we achieve this:

• Add or subtract terms to both sides of the equation to eliminate constants from the variable term. For example, add 8 to both sides to remove \(-8\): \(3x - 8 + 8 = 25 + 8\), simplifying gives \(3x = 33\).
• Next, eliminate coefficients by dividing both sides of the equation by the coefficient. For \(3x = 33\), divide by 3 to solve for \(x\): \(x = \frac{33}{3}\), simplifying gives \(x = 11\).

Through these steps, you systematically strip away layers to expose the variable, resulting in a clear and comprehensible solution.

Verification of Solutions

Once you've found a solution for the variable, it’s important to verify that this solution is correct. Verification ensures that no errors were made during the manipulation process and that the solution satisfies the original equation.
For our exercise, we found \(x = 11\) as a solution. To verify:

• Substitute \(x = 11\) back into the original equation: \(3(11) - 8 = 25\).
• Calculate the left side of the equation: \(33 - 8 = 25\).
• Check that both sides of the equation are equal. Since they are, \(x = 11\) is indeed correct.

Verification is a simple yet powerful tool that provides confidence in the solution and reinforces your understanding of the problem-solving process.