- Distribute constants on both sides to get \(-12x - 2x - 18 = 5x + 20\).
- Simplify the left side to \(-14x - 18 = 5x + 20\).
- Isolate \(x\) terms on the right side and constants on the left to obtain \(-38 = 19x\).
- Solve for \(x\) to find the final result \(\boxed{-2}\).
Explanation
- Analyze the Equation
The problem asks to find the value of \(x\) that satisfies the equation:
\[-12x - 2(x + 9) = 5(x + 4)\]
This is a linear equation in one variable. To find the solution, we need to isolate the variable \(x\) on one side of the equality. We will start by expanding the terms within the parentheses.
- Simplify the Expression
Distribute the constant terms across the parentheses on both sides of the equation:
\[-12x - 2(x) - 2(9) = 5(x) + 5(4)\]
\[-12x - 2x - 18 = 5x + 20\]
Now, combine the like terms on the left-hand side, which are \(-12x\) and \(-2x\):
\[-14x - 18 = 5x + 20\]
- Isolate the Variable \(x\)
Move all terms containing \(x\) to one side of the equation and all constant terms to the other side. Add \(14x\) to both sides and subtract \(20\) from both sides:
\[-18 - 20 = 5x + 14x\]
Simplifying both sides gives:
\[-38 = 19x\]
- Solve and Verify
Finally, divide both sides of the equation by \(19\) to solve for \(x\):
\[x = \frac{-38}{19}\]
\[x = -2\]
We can verify this result by substituting \(x = -2\) back into the original equation:
Left-hand side: \(-12(-2) - 2(-2 + 9) = 24 - 2(7) = 24 - 14 = 10\)
Right-hand side: \(5(-2 + 4) = 5(2) = 10\)
Since both sides equal \(10\), our solution is correct. The correct option is \(-2\).
Examples
Linear equations are essential in many real-world scenarios, such as budgeting or predicting costs. For example, if a company charges a base fee for equipment plus an hourly rate, you can set up a linear equation to determine the number of hours you can use the equipment within a fixed budget. By solving for the unknown number of hours, you make informed decisions about resource management, just like how we solved for \(x\) in our equation.