- Distribute the constant 6 across the parentheses to rewrite the equation as \(6x - 6 - 3 = 9\).
- Simplify by combining the constants on the left side, resulting in \(6x - 9 = 9\).
- Isolate the variable term by adding 9 to both sides to get \(6x = 18\).
- Divide by 6 to find the solution: \(x = \boxed{3}\).
Explanation
- Data and Problem Analysis
We are tasked with solving the linear equation \(6(x - 1) - 3 = 9\) for the variable \(x\). This equation involves a single variable and constant terms. Our objective is to isolate \(x\) by systematically simplifying the expression, which involves distribution, combining like terms, and applying inverse operations to eliminate constants and coefficients.
- Distributing to Remove Parentheses
First, we apply the distributive property to the expression \(6(x - 1)\). Multiplying 6 by each term inside the parentheses, we obtain \(6x - 6\). Substituting this back into the original equation, we get:
\[6x - 6 - 3 = 9\]
- Combining Like Terms
Next, we simplify the left side of the equation by combining the constant terms \(-6\) and \(-3\). Since \(-6 - 3 = -9\), the equation simplifies to:
\[6x - 9 = 9\]
- Undoing Addition or Subtraction
To isolate the term with \(x\), we need to undo the subtraction of 9. We add 9 to both sides of the equation:
\[6x - 9 + 9 = 9 + 9\]
\[6x = 18\]
- Undoing Multiplication and Finding the Final Answer
Finally, we solve for \(x\) by undoing the multiplication. We divide both sides of the equation by the coefficient of \(x\), which is 6:
\[\frac{6x}{6} = \frac{18}{6}\]
\[x = 3\]
Thus, the solution to the equation is \(x = 3\).
Examples
Solving linear equations is a fundamental skill used in many real-world budgeting and resource allocation tasks. For instance, if you are buying items for a project where you pay a fixed fee for shipping and a certain amount per unit, you can set up a linear equation to determine how many units you can afford within a fixed budget. By representing the total cost as a function of the number of items and solving for that number, you ensure your spending remains within your financial limits. This algebraic approach turns complex resource decisions into clear, manageable calculations.