Verified Answer :

  • Expand the given equation: \(at + az = 45z + 67\).
  • Group the terms involving \(z\) on one side: \(az - 45z = 67 - at\).
  • Factor \(z\) out: \(z(a - 45) = 67 - at\).
  • Divide by \((a - 45)\) to isolate \(z\), giving the final answer: \(\boxed{\frac{67 - at}{a - 45}}\).

Explanation

  1. Understanding the Problem and Variables

The problem asks us to solve for the variable \(z\) in the linear equation \(a(t + z) = 45z + 67\). Here, \(a\) and \(t\) are treated as constants (parameters). To isolate \(z\), we need to manipulate the algebraic expression using properties of equality. We assume \(a eq 45\) to ensure the equation remains well-defined.

  1. Expanding the Equation

First, we expand the left side of the equation:
\[a(t + z) = at + az\]
Now, substitute this back into the original equation:
\[at + az = 45z + 67\]

  1. Isolating the z-Terms

To group all terms containing \(z\) on one side, subtract \(45z\) from both sides and subtract \(at\) from both sides:
\[az - 45z = 67 - at\]
Next, factor out the variable \(z\) on the left side of the equation:
\[z(a - 45) = 67 - at\]

  1. Solving for z

Finally, we solve for \(z\) by dividing both sides of the equation by \((a - 45)\):
\[z = \frac{67 - at}{a - 45}\]
This expression gives the value of \(z\) in terms of \(a\) and \(t\), assuming \(a eq 45\).

Examples

Algebraic manipulation is fundamental in physical sciences and engineering. For instance, if you are calculating the final position of an object moving with a constant acceleration where its initial velocity is affected by a temperature-dependent parameter, you might end up with an equation where you need to isolate the time variable or the distance variable. By rearranging equations as we did here, you can easily adapt your models to predict different variables depending on what data you already have.