Verified Answer :

  • Distribute the constant 3 to get \(-12x - 15 + 5x + 4 = -4\).
  • Combine like terms to simplify the expression to \(-7x - 11 = -4\).
  • Add 11 to both sides to get \(-7x = 7\).
  • Divide by -7 to find the final result: \(\boxed{-1}\).

Explanation

  1. Problem Analysis

The problem asks to solve for \(x\) in the linear equation \(3(-4x - 5) + 5x + 4 = -4\). To isolate the variable \(x\), we need to simplify both sides of the equation by performing algebraic operations such as distribution, combining like terms, and isolating the variable.

  1. Distributing the Constant

We begin by applying the distributive property to the first term, \(3(-4x - 5)\). Multiplying 3 by both terms inside the parentheses gives \(3 \times (-4x) + 3 \times (-5) = -12x - 15\). Substituting this back into the original equation, we get:
\[-12x - 15 + 5x + 4 = -4\]

  1. Combining Like Terms

Next, we group the like terms on the left-hand side of the equation. We combine the \(x\)-terms (\(-12x\) and \(5x\)) and the constant terms (\(-15\) and \(4\)):
\[(-12x + 5x) + (-15 + 4) = -4\]
\[-7x - 11 = -4\]

  1. Isolating the Variable Term

Now, we isolate the \(x\)-term by adding 11 to both sides of the equation:
\[-7x - 11 + 11 = -4 + 11\]
\[-7x = 7\]

  1. Solving for x

Finally, we solve for \(x\) by dividing both sides of the equation by -7:
\[x = \frac{7}{-7}\]
\[x = -1\]
Therefore, the solution to the equation is \(x = -1\).

Examples

Linear equations are fundamental in modeling various real-world situations, such as calculating costs. For instance, if a service provider charges a fixed fee and an hourly rate, you can model the total cost using a linear equation. If you have a total budget and know the fixed fee and hourly rate, you can solve for the number of hours (\(x\)) you can afford, just like solving the equation above helps find an unknown value based on provided conditions.

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