Verified Answer :

  • Distribute the \(-3\) across the parentheses to get \(2 = -3z - 21 - 16\).
  • Combine the constants \(-21\) and \(-16\) to simplify the right side to \(2 = -3z - 37\).
  • Add \(37\) to both sides to isolate the \(z\) term, resulting in \(39 = -3z\).
  • Divide by \(-3\) to solve for \(z\), yielding \(z = \boxed{-13}\).

Explanation

  1. Problem Analysis

We are given the linear equation \(2 = -3(z + 7) - 16\). Our goal is to isolate the variable \(z\) and find its value. This involves distributing, simplifying, and then isolating the variable through inverse operations.

  1. Distributing the Constant

First, we distribute the \(-3\) into the expression \((z + 7)\):
\[2 = -3 \times z + (-3) \times 7 - 16\]
\[2 = -3z - 21 - 16\]

  1. Combining Constants

Next, we combine the constant terms on the right side of the equation:
\[-21 - 16 = -37\]
So the equation becomes:
\[2 = -3z - 37\]

  1. Isolating the Variable Term

Now, we isolate the variable term \(-3z\) by adding \(37\) to both sides of the equation:
\[2 + 37 = -3z\]
\[39 = -3z\]

  1. Solving for z

Finally, we solve for \(z\) by dividing both sides by \(-3\):
\[z = \frac{39}{-3} = -13\]
Thus, the value of \(z\) is \(-13\).

Examples

Linear equations are used in countless everyday situations. For example, if you are saving money for a trip, you might start with a certain amount and save a fixed weekly amount to reach a goal. If the equation represents your total balance, finding the unknown time period needed to reach a specific goal is a direct application of solving a simple linear equation like the one we just solved. This fundamental skill forms the basis for algebra and more complex problem-solving in finance, physics, and engineering.

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