91Ó°ÊÓ

Equation manipulation is a fundamental skill in algebra that lets us change the form of equations to make them easier to solve. In this problem, we started with an equation given as: \[ S = \frac{k A}{P} \] which involves the variables \(S\), \(A\), \(P\), and \(k\). To find \(k\), we needed to rewrite the equation with \(k\) by itself. This process involved rearranging the original equation. By multiplying both sides by \(P\), you can remove the division, resulting in: \[ S \cdot P = k \cdot A \] To further isolate \(k\), one more step of dividing both sides by \(A\) is necessary; this leads to: \[ k = \frac{S \cdot P}{A} \] This manipulation allows you to clearly see that \(k\) can be directly solved when you know \(S\), \(P\), and \(A\). Remember: you can manipulate equations as long as you perform the same operations on both sides. That maintains equality.

Isolating a variable means getting it by itself on one side of the equation. This concept is crucial for solving equations because it helps to find the value of unknown quantities. In our exercise, the goal was to find \(k\), so we needed to get \(k\) alone. The original equation was \(S = \frac{k A}{P}\). Our task was to isolate \(k\) by performing algebraic operations that preserve the equation's balance. Here's what to do:

• First, get rid of the fraction by multiplying both sides by \(P\): \(S \cdot P = k \cdot A\).
• Next, divide both sides by \(A\) to solve for \(k\): \(k = \frac{S \cdot P}{A}\).

Now \(k\) is isolated and ready for substitution. Isolating helps to focus on one variable, making the problem simpler and clearer.

The substitution method involves replacing variables with given values to simplify and solve equations. It is a vital strategy, especially in problems where you have concrete numbers to work with. Once \(k\) was isolated as \(k = \frac{S \cdot P}{A}\), we applied the substitution method.We identified the given values from the problem:

• \(S = 12,000\)
• \(P = 40\)
• \(A = 60,000\)

Substituting these into the formula gives us:\[ k = \frac{12,000 \cdot 40}{60,000} \]By calculating this expression, multiplying and dividing as needed, we found the solution: \(k = 8\).Substitution replaces abstract variables with concrete numbers, simplifying the solution process and making your answer computation straightforward.