Verified Answer :

  • Divide both sides of the equation \(5y = c(6 + a)\) by \(c\) to get \(\frac{5y}{c} = 6 + a\).
  • Subtract 6 from both sides to isolate \(a\): \(a = \frac{5y}{c} - 6\).
  • Optionally, express as a single fraction: \(a = \frac{5y - 6c}{c}\).
  • Final answer: \(a = \boxed{\frac{5y - 6c}{c}}\).

Explanation

  1. Analyzing the Given Equation

The goal is to solve the linear equation \(5y = c(6 + a)\) for the variable \(a\). We assume that \(c eq 0\) so that division by \(c\) is well-defined. Our objective is to isolate \(a\) on one side of the equation.

  1. Isolating the Term with \(a\)

Start with the original equation:
\[5y = c(6 + a)\]
To isolate the term containing \(a\), divide both sides of the equation by \(c\):
\[\frac{5y}{c} = 6 + a\]

  1. Solving for \(a\)

Now, isolate \(a\) by subtracting \(6\) from both sides:
\[a = \frac{5y}{c} - 6\]
To write this as a single fraction, express 6 as \(\frac{6c}{c}\):
\[a = \frac{5y}{c} - \frac{6c}{c}\]
\[a = \frac{5y - 6c}{c}\]
This expression gives us the value of \(a\) in terms of \(y\) and \(c\).

Examples

Imagine you are running a small shop and you need to calculate the cost per item. If your total revenue \(5y\) is related to the number of items sold \(c\) and the cost adjustment factor \(6 + a\), you can use this formula to determine the specific adjustment value \(a\) needed to match your target revenue for a given number of units sold. Understanding how to isolate variables allows you to adapt formulas to solve for whichever component of your business model is currently unknown.

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