Make $a$ the subject of the formula where $a$ is positive:
$a+2 b=6 a c+5 d$
We are given the equation \(a + 2b = 6ac + 5d\) and we want to isolate the variable \(a\). This involves collecting all terms involving \(a\) on one side of the equality and all other terms on the other side. Once isolated, we can use algebraic manipulation to express \(a\) in terms of the other variables \(b, c, \text{ and } d\). We are also given the condition \(a > 0\).
Start with the original equation:
\[a + 2b = 6ac + 5d\]
To isolate the terms with \(a\), subtract \(6ac\) and \(2b\) from both sides:
\[a - 6ac = 5d - 2b\]
Next, factor out \(a\) from the left side:
\[a(1 - 6c) = 5d - 2b\]
Finally, divide both sides by \((1 - 6c)\) to solve for \(a\):
\[a = \frac{5d - 2b}{1 - 6c}\]
The expression for \(a\) can also be rewritten by multiplying both the numerator and the denominator by \(-1\):
\[a = \frac{-(5d - 2b)}{-(1 - 6c)} = \frac{2b - 5d}{6c - 1}\]
This form is equivalent to the previous one and often preferred. Given the condition \(a > 0\), it must hold that \(\frac{2b - 5d}{6c - 1} > 0\). Thus, the final expression for \(a\) is \(\boxed{\frac{2b - 5d}{6c - 1}}\).
Rearranging formulas is a fundamental skill in science and engineering. For instance, consider the ideal gas law \(PV = nRT\). If you needed to solve for the temperature \(T\), you would divide both sides by \(nR\) to get \(T = \frac{PV}{nR}\). Similarly, the technique of factoring and isolating variables allows you to understand how different physical quantities, such as pressure, volume, and temperature, depend on each other, which is essential for conducting experiments and solving real-world problems.