- Group the like terms: \((16 + 5)x - 4 = -67\), which simplifies to \(21x - 4 = -67\).
- Isolate the term with \(x\) by adding \(4\) to both sides: \(21x = -63\).
- Divide by the coefficient \(21\) to solve for \(x\): \(x = -3\).
- The final result is \(\boxed{-3}\).
Explanation
- Analyzing the Linear Equation
The equation to solve is \(16x - 4 + 5x = -67\). This is a linear equation in one variable \(x\). Our goal is to isolate \(x\) to find its value. The steps involved include grouping like terms, isolating the variable term, and then solving for \(x\).
- Combining Like Terms
First, combine the terms containing \(x\) on the left side of the equation:
\[(16x + 5x) - 4 = -67\]
\[21x - 4 = -67\]
- Isolating the Term with x
Next, isolate the term with \(x\) by adding \(4\) to both sides of the equation:
\[21x = -67 + 4\]
\[21x = -63\]
- Solving for x
Finally, solve for \(x\) by dividing both sides of the equation by the coefficient of \(x\), which is \(21\):
\[x = \frac{-63}{21}\]
\[x = -3\]
- Conclusion and Final Answer
By solving the equation \(16x - 4 + 5x = -67\), we determined that \(x = -3\). Comparing this to the given options:
- \(x = \frac{1}{3}\)
- \(x = -3\)
- \(x = 3\)
- \(x = -\frac{1}{3}\)
The correct option is \(x = -3\). Thus, the solution is \(\boxed{-3}\).
Examples
Linear equations appear frequently in budgeting and daily calculations. For instance, if you have a base fee of \(\$4\) for a service and pay \(\$16\) for every hour, plus an additional \(\$5\) per hour due to a surcharge, the total cost for \(x\) hours might be modeled by \(16x + 5x - 4\). If your total bill came to \(\$67\) less than what you expected (or if you were given a credit that results in \(-67\)), you would set up and solve the equation \(16x + 5x - 4 = -67\) to find the unknown number of hours. Understanding how to solve these equations helps in managing finances and predicting outcomes.