- Apply the distributive property: \(2x + 6 + 4 = 8\).
- Simplify by combining like terms: \(2x + 10 = 8\).
- Isolate the variable term: \(2x = -2\).
- Solve for \(x\) by dividing: \(x = -1\).
Final answer: \(\boxed{-1}\)
Explanation
- Problem Analysis
The goal is to solve the linear equation \(2(x+3)+4=8\) by applying systematic algebraic operations. We will follow the order of operations: expand the expression using the distributive property, simplify by combining like terms, and then isolate the variable \(x\) using inverse operations.
- Distributing the Term
First, apply the distributive property to the expression \(2(x+3)\). Multiplying 2 by both \(x\) and 3 inside the parentheses yields:
\[2x + 6 + 4 = 8.\]
- Combining Like Terms
Next, combine the constant terms \(6\) and \(4\) on the left side of the equation:
\[2x + 10 = 8.\]
- Undoing Addition
Now, isolate the term containing \(x\) by subtracting \(10\) from both sides of the equation:
\[2x = 8 - 10\]
\[2x = -2.\]
- Undoing Multiplication and Finalizing the Answer
Finally, solve for \(x\) by dividing both sides of the equation by \(2\):
\[x = \frac{-2}{2}\]
\[x = -1.\]
Thus, the value of \(x\) that satisfies the equation is \(-1\).
Examples
Linear equations are essential in many real-life situations involving budgets or resource allocation. For example, if you are planning a party with a base cost and a cost per guest, you can use a linear equation to find the number of guests you can afford. If the total budget for a party is \$100, the venue rental is \$40, and each guest costs \$15 for food, the equation would be \(40 + 15x = 100\). Solving this as we did above allows you to determine exactly how many guests (\(x\)) you can invite while staying within your budget.