91Ó°ÊÓ

Isolating variables is a crucial step in solving equations, especially when you need to express one variable in terms of others. The goal is to "isolate" the variable you're solving for. This means you want to manipulate the equation such that the variable is alone on one side of the equation while all other terms are on the opposite side.
For the equation given, which is \(d = rt\), our task is to isolate \(t\). Since \(t\) is currently multiplied by \(r\), we perform the inverse operation to isolate it by dividing both sides of the equation by \(r\).
This results in \(\frac{d}{r} = \frac{rt}{r}\). With this manipulation, you effectively remove \(r\) from the equation, leaving you with \(t\) by itself on one side. This step is foundational in equation solving and applies to many different kinds of problems, not just this one.

Algebraic manipulation involves using mathematical operations to transform an equation or expression to reach a desired form. The operations can include addition, subtraction, multiplication, division, and factorization, among others. In our case, with the equation \(d = rt\), we're focusing on division to manipulate the equation.
By dividing both sides of the equation by \(r\), we apply the principle that what's done to one side of an equation must be done to the other. This balances the equation and helps in solving it without altering its equality. After performing the division, you get \(\frac{d}{r} = t\), demonstrating how division, a basic algebraic operation, has simplified our problem to express \(t\) in terms of \(d\) and \(r\).
Understanding and efficiently applying algebraic manipulation is key to effectively problem-solving in algebra and beyond.

A variable expression is a mathematical phrase that can include numbers, variables, and operational signs. In the given problem, we want to provide an expression for \(t\) that relies on the other given variables, \(d\) and \(r\).
The expression \(t = \frac{d}{r}\) is what we end up with after isolating \(t\) through algebraic manipulation. This expression shows how \(t\) relates to the other variables in the context of the given equation \(d = rt\). This understanding allows for quick recalculations or evaluations if values for \(d\) and \(r\) are known.
Grasping how to form and interpret variable expressions is vital, as it offers insight into relationships represented by equations and assists in further mathematical explorations and applications.