Problem 14 - College Algebra with Modeling and Visualization

Question

Solve the equation and check your answer. $$ 4 x-8=0 $$

Step-by-Step Solution

Answer

The solution is \(x = 2\), verified by substitution.

Step 1: Identify the Equation

The equation given is \(4x - 8 = 0\). This is a linear equation in one variable, where \(x\) is the variable.

Step 2: Isolate the Variable Term

To isolate \(4x\), we add 8 to both sides of the equation: \\[4x - 8 + 8 = 0 + 8\] \This simplifies to: \\[4x = 8\]

Step 3: Solve for x

Now, divide both sides by 4 to solve for \(x\): \\[\frac{4x}{4} = \frac{8}{4}\] \This simplifies to: \\[x = 2\]

Step 4: Check the Solution

Let's substitute \(x = 2\) back into the original equation to verify: \\[4(2) - 8 = 0\] \Calculate \(4(2) = 8\), so \\[8 - 8 = 0\] \This is true, confirming the solution is correct.

Key Concepts

Equation Solving

Equation solving is the process of finding the value of the variable that makes the equation true. In the provided exercise, we're dealing with the equation:\[4x - 8 = 0\]To solve this, you need to find the value of \(x\) that satisfies the equation. The key idea is to perform operations that simplify or rearrange the equation, ultimately isolating the variable on one side.

• Begin by identifying the structure of the equation. Here, \(4x - 8\) is a linear expression.
• Linear equations typically follow the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.
• The goal is to manipulate the equation so that \(x\) appears on one side and the constant term is isolated on the other.

This simple equation can be tackled using straightforward arithmetic operations to reveal the solution.

Algebraic Manipulation

Algebraic manipulation allows us to rearrange equations to find the solution. Let's see how this works with our equation:\[4x - 8 = 0\]First, you need to eliminate the constant term from the left side. You do this by applying inverse operations, like addition or subtraction.**Step 1: Eliminate the Constant** Add 8 to both sides:\[4x - 8 + 8 = 0 + 8\]to simplify to:\[4x = 8\]**Step 2: Isolate the Variable** Next, divide both sides by the coefficient of \(x\), which is 4, to isolate \(x\):\[\frac{4x}{4} = \frac{8}{4}\]which simplifies to:\[x = 2\]By performing these operations, you've successfully isolated \(x\), finding the solution to the equation.

Verification of Solutions

Verifying a solution ensures that the calculated value satisfies the original equation. It is a crucial step to confirm accuracy in equation solving. Let's verify the solution found for the equation \(4x - 8 = 0\).**Substitution** Replace the variable \(x\) in the original equation with the solution you have found, which here is 2.\[4(2) - 8 = 0\]**Perform the Calculation** Calculate the left side:\[4 \times 2 = 8\]Then,\[8 - 8 = 0\]**Conclusion** Since both sides of the equation equal zero, the verification process confirms that \(x = 2\) is indeed the correct solution to the equation. Ensuring the correctness of a solution through verification helps solidify understanding and secures confidence in problem-solving skills.