Verified Answer :

  • Start with the original equation: \(4x - 8 = 12\).
  • Add 8 to both sides to isolate the term with \(x\): \(4x = 20\).
  • Divide both sides by 4 to solve for \(x\): \(x = 5\).
  • The solution is \(\boxed{5}\).

Explanation

  1. Problem Analysis

The objective is to solve for the variable \(x\) in the linear equation \(4x - 8 = 12\). A linear equation expresses a relationship where the variable is raised to the power of 1. To solve it, we need to isolate \(x\) on one side of the equation. This involves applying inverse operations to both sides to maintain the equality. We start by moving the constant term to the other side and then isolating \(x\) by removing its coefficient.

  1. Solving the Equation

We begin with the original equation:
\[4x - 8 = 12\]
To isolate the term with \(x\), we add 8 to both sides of the equation:
\[4x - 8 + 8 = 12 + 8\]
\[4x = 20\]
Now, to solve for \(x\), we divide both sides by the coefficient of \(x\), which is 4:
\[\frac{4x}{4} = \frac{20}{4}\]
\[x = 5\]

  1. Conclusion

The value of the variable \(x\) that satisfies the equation \(4x - 8 = 12\) is 5. We can verify this by substituting \(x = 5\) back into the original equation: \(4(5) - 8 = 20 - 8 = 12\). The equation holds true, so the solution is correct. The final answer is \(\boxed{5}\).

Examples

Solving linear equations is a fundamental skill used in everyday life, such as when calculating the final price of an item during a sale. Suppose you have a budget of \(\$12\) for a school project, and you have already spent \(\$8\) on a poster board. You want to know how many identical packs of stickers costing \(\$4\) each you can buy with the remaining money. By setting up the equation \(4x + 8 = 12\), you can determine the number of packs you can afford. In our specific case, \(4x = 4\), so \(x = 1\) pack of stickers. This process of isolating variables helps in budgeting, planning schedules, and understanding proportional costs.