- Distribute 2 into the parentheses: \(6x + 8 + 5 = 23\).
- Combine the constant terms: \(6x + 13 = 23\).
- Subtract 13 from both sides: \(6x = 10\).
- Divide by 6 and simplify the resulting fraction: \(x = \boxed{\frac{5}{3}}\).
Explanation
- Problem Analysis
We are tasked with solving the linear equation \(2(3x + 4) + 5 = 23\) for the variable \(x\). This equation involves a distribution step, combining constant terms, and applying inverse operations to isolate \(x\). Following a structured approach will ensure accuracy in finding the value of \(x\).
- Simplification and Distribution
First, we apply the distributive property to the expression \(2(3x + 4)\). Multiplying 2 by both \(3x\) and 4 gives:
\[6x + 8 + 5 = 23\]
Next, we combine the constant terms on the left side, which are 8 and 5:
\[6x + (8 + 5) = 23\]
\[6x + 13 = 23\]
- Isolating the Variable Term
To isolate the term with \(x\), we subtract 13 from both sides of the equation:
\[6x = 23 - 13\]
\[6x = 10\]
- Solving for x
Finally, we solve for \(x\) by dividing both sides of the equation by 6:
\[x = \frac{10}{6}\]
By simplifying the fraction \(\frac{10}{6}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2, we obtain:
\[x = \frac{5}{3}\]
Thus, the solution to the equation is \(x = \frac{5}{3}\).
Examples
Solving linear equations like this one is foundational in algebra and appears frequently in real-world scenarios. For example, if a taxi service charges a base fee of \(\$5\) plus \(\$2\) for every mile driven, and you have \(\$23\) to spend, you can model the distance \(x\) you can travel using the equation \(2x + 5 = 23\). Mastering these algebraic steps allows you to quickly solve for unknown quantities in various financial, scientific, or everyday life situations.