Verified Answer :

  • Start with the original equation: \(19x + rx = -37x + w\).
  • Move all terms involving \(x\) to the left side: \(19x + rx + 37x = w\).
  • Factor out \(x\) to get \((56 + r)x = w\).
  • Solve for \(x\) by dividing by \((56 + r)\) to obtain \(x = \boxed{\frac{w}{56 + r}}\).

Explanation

  1. Analysis of the Equation

We are given a linear equation in terms of \(x\):
\[19x + rx = -37x + w\]
Our goal is to solve for \(x\). We treat \(r\) and \(w\) as constants or parameters. The standard procedure for solving linear equations is to isolate the variable of interest, in this case \(x\), on one side of the equation.

  1. Isolating \(x\)

First, we move all terms containing \(x\) to the left-hand side of the equation. We can achieve this by adding \(37x\) to both sides of the equation:
\[19x + rx + 37x = w\]
Now, group the terms that have \(x\) as a common factor:
\[(19 + r + 37)x = w\]
Simplifying the sum inside the parentheses, \(19 + 37 = 56\), so the equation becomes:
\[(56 + r)x = w\]

  1. Finalizing the Solution

To solve for \(x\), we divide both sides by the coefficient \((56 + r)\). We assume that \(56 + r eq 0\), which is necessary for \(x\) to be defined as a finite value.
\[x = \frac{w}{56 + r}\]
Thus, the expression for \(x\) is \(\frac{w}{56+r}\).

Examples

In engineering, this type of equation is common when balancing forces or calculating resistances in a circuit. For example, if you are modeling the total resistance \(R_{total}\) of a circuit where some components depend on a variable factor \(x\), such as \(19x + rx = -37x + w\), solving for \(x\) allows you to determine the necessary input or variable adjustment \(x\) to achieve a desired output or work load \(w\). Understanding how to isolate variables helps in designing systems where components have interrelated properties.

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