Problem 6 - Intermediate Algebra

Question

Solve \(d=r t\) for \(t\)

Step-by-Step Solution

Answer

The solution is \(t = \frac{d}{r}\).

Step 1: Understand the Equation

The given equation is \(d = rt\), where \(d\) is the distance, \(r\) is the rate (or speed), and \(t\) is the time. We want to solve this equation for \(t\).

Step 2: Isolate \(t\)

To solve for \(t\), we need to isolate it on one side of the equation. Currently, \(t\) is multiplied by \(r\). To isolate \(t\), we divide both sides of the equation by \(r\). The equation becomes:\[\frac{d}{r} = t\]

Step 3: Verify the Solution

Now that \(t\) is isolated, the solution is \(t = \frac{d}{r}\). To verify this, we can check that multiplying both sides by \(r\) reverts back to the original equation:\[r \times \left( \frac{d}{r} \right) = d\]This simplifies to \(d = d\), confirming our solution is correct.

Key Concepts

Solving Equations

Equations are like puzzles where you find the missing piece, and in mathematics, that's often a variable. Solving equations means determining the value or expression for a variable that makes the equation true. In the equation we dealt with, \(d = rt\), the aim is to find the expression that defines \(t\) in terms of \(d\) and \(r\).
The solution involves breaking down the components of the equation using techniques that maintain equality.When solving equations:

• Maintain balance by doing the same operation on both sides.
• Identify the operation involving the variable you want to solve for.
• Apply inverse operations to isolate the variable.

For instance, if a variable is multiplied or divided, use division or multiplication respectively to isolate the variable on one side. Remember, equations are just a balance scale, and every action on one side should be matched on the other to keep everything equal.

Distance-Rate-Time Formula

The Distance-Rate-Time formula is a fundamental principle used in solving problems related to motion. The formula is written as \(d = rt\), where \(d\) represents distance, \(r\) is the rate or speed, and \(t\) is time.
This formula tells us how these three components interact with each other:

• Distance \(d\) is how far an object travels.
• Rate \(r\) is the speed or velocity at which the object moves.
• Time \(t\) is the duration over which the object has been moving.

Using this relationship, you can solve for any of the three variables if you know the other two. For example, if you know the rate and time, you can find the distance. However, if distance and rate are given, finding time becomes straightforward.
Understanding this formula is crucial in algebra as it applies in scenarios from calculating travel time to predicting movement based on speed, helping make informed decisions in real-world applications.

Variable Isolation

In algebra, isolating a variable means rearranging an equation in such a way that the variable you're interested in stands alone on one side. This concept is crucial for solving equations effectively.
To isolate a variable, you need to reverse the operations connected to the variable. In the equation \(d = rt\), to isolate \(t\), consider:

• Identify how \(t\) is connected to other elements, in this case, \(t\) is multiplied by \(r\).
• Use the operation that reverses this connection — here, division. By dividing both sides by \(r\), \(t\) is isolated.
• The equation becomes \(t = \frac{d}{r}\), effectively separating \(t\) from other variables.

Isolating variables is important since it allows you to express one variable in terms of others, thus simplifying the complexity involved in solving mathematical equations. This method is extensively used in various fields like physics, engineering, and finance, wherever mathematical equations are involved.