Verified Answer :

  • Distribute the constant 3 into the parentheses to get \(-9x - 12 + 3x - 1 = -1\).
  • Combine like terms to simplify the equation to \(-6x - 13 = -1\).
  • Add 13 to both sides to isolate the variable term, resulting in \(-6x = 12\).
  • Divide by -6 to find the solution: \(x = -2\).
  • Final Answer: \(\boxed{-2}\)

Explanation

  1. Problem Analysis

We are given the linear equation \[3(-3x - 4) + 3x - 1 = -1.\] Our objective is to find the value of \(x\) that satisfies this equality. We will approach this by expanding the parentheses, combining like terms, and isolating the variable \(x\).

  1. Expanding the Parentheses

First, we expand the expression \(3(-3x - 4)\) by distributing the \(3\) to each term inside the parentheses:
\[3 \times (-3x) + 3 \times (-4) = -9x - 12.\]
Substituting this back into our original equation, we get:
\[-9x - 12 + 3x - 1 = -1.\]

  1. Combining Like Terms

Next, we identify and combine the like terms on the left-hand side of the equation.
The terms containing \(x\) are \(-9x\) and \(3x\), which sum to \(-9x + 3x = -6x\).
The constant terms are \(-12\) and \(-1\), which sum to \(-12 - 1 = -13\).
So, the equation simplifies to:
\[-6x - 13 = -1.\]

  1. Isolating the Variable and Solving

Now, we isolate the \(x\) term by adding \(13\) to both sides of the equation:
\[-6x - 13 + 13 = -1 + 13\]
\[-6x = 12.\]
To solve for \(x\), we divide both sides by \(-6\):
\[x = \frac{12}{-6} = -2.\]

Examples

Solving linear equations is a fundamental skill used in many practical scenarios. For instance, imagine you are budgeting for a project. If you have a total budget that depends on the number of hours worked or units produced, you often end up with a linear equation where you need to solve for an unknown quantity, such as the maximum number of hours you can afford. Mastering these techniques allows for precise financial planning and resource management in everyday life.

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