- Distribute the \(2\) inside the parentheses: \(4x + 6 - 6 = 16\).
- Combine the constant terms: \(4x = 16\).
- Divide both sides by \(4\) to solve for \(x\): \(x = 4\).
- The solution to the equation is \(\boxed{4}\).
Explanation
- Problem Analysis
The given equation is \(2(2x + 3) - 6 = 16\). Our goal is to isolate the variable \(x\). This equation is linear, involving a single variable, and can be solved systematically using algebraic techniques. We will follow the order of operations by first eliminating the parentheses through distribution, then simplifying constant terms, and finally isolating the variable.
- Solving the Equation
First, we distribute the constant \(2\) to the terms inside the parentheses:
\[2 \times 2x + 2 \times 3 - 6 = 16\]
\[4x + 6 - 6 = 16\]
Next, we combine the constant terms on the left side:
\[4x + 0 = 16\]
\[4x = 16\]
Now that the equation is simplified to \(4x = 16\), we isolate \(x\) by dividing both sides of the equation by \(4\):
\[x = \frac{16}{4}\]
\[x = 4\]
Finally, to ensure the result is correct, we can substitute \(x=4\) back into the original equation:
\[2(2(4) + 3) - 6 = 2(8 + 3) - 6 = 2(11) - 6 = 22 - 6 = 16\]
Since the substitution satisfies the original equation, the value \(x=4\) is correct.
Examples
Linear equations are fundamental in everyday scenarios where a fixed rate is applied to an unknown quantity. For instance, if you are planning a trip and need to rent a car, the cost might be calculated as a base fee plus a daily rate. If the total cost for 4 days is \$160 and you know the daily rate is \$30, you can model this as \(4 \times 30 + \text{Base Fee} = 160\), helping you determine the initial fixed cost of renting the car. Understanding these algebraic steps allows you to solve for unknown variables in various real-world budgeting and planning situations.