Problem 2 - Algebra A Combined Function

Question

Substitute the given values into each given formula and solve for the unknown variable. $$ d=r t ; \quad d=195, t=3 $$

Step-by-Step Solution

Answer

The rate \(r\) is 65.

Step 1: Identify the formula and known values

The formula provided is \[ d = rt \]where \(d\) is the distance, \(r\) is the rate (or speed), and \(t\) is the time. Here, \(d = 195\) and \(t = 3\). Our goal is to find the value of \(r\).

Step 2: Rearrange the formula to solve for the unknown variable

Since we need to find \(r\), we can rearrange the formula for \(r\). Starting from \[ d = r t \]we divide both sides by \(t\) to get the equation for \(r\):\[ r = \frac{d}{t} \]

Step 3: Substitute known values into the rearranged formula

Now, substitute the known values of \(d\) and \(t\) into the equation:\[ r = \frac{195}{3} \]

Step 4: Calculate the value of the unknown variable

Perform the division to find \(r\):\[ r = \frac{195}{3} = 65 \] Thus, the rate \(r\) is 65.

Key Concepts

Substitution Method

The substitution method is a technique used in mathematics to make problem-solving more manageable.
In this context, it involves plugging in known values into a formula or equation to find an unknown variable.
By substituting values into the equation, it becomes easier to work through the calculations.In our exercise, we begin with the known equation \[ d = rt \] where we are given some values like \(d = 195\) and \(t = 3\). By recognizing which variable is unknown, we focus on solving for that.
The known values are directly substituted for their respective variables in the formula.This method not only simplifies the equation but also makes it clear and straightforward to solve.
It's a fundamental technique especially useful in algebra and physics to determine unknowns in equations.

Distance Formula

The distance formula, represented as \( d = rt \), is used to calculate the distance traveled by an object moving at a constant speed over a period of time.
This formula is part of the broader study of motion and is essential for solving real-world problems involving speed and distance.

• \(d\) stands for distance, which refers to the total path covered by the moving object.
• \(r\) stands for rate or speed, depicting how fast the object is moving.
• \(t\) stands for time, which illustrates the duration of the object's travel.

When you're provided with any two of these variables, you can easily figure out the third. In this exercise, knowing the distance and time allowed us to find the rate of speed.

Solving for Unknown Variable

When tackling problems that require solving for an unknown variable, the key step is to isolate the variable you need to find.
In our exercise with \(d = rt\), the unknown variable is the rate \(r\). To solve for \(r\), we rearrange the formula so \(r\) stands alone: \[ r = \frac{d}{t} \] By dividing both sides of the equation by \(t\), we effectively remove \(t\) from the right side, isolating \(r\).Once we have the equation \( r = \frac{d}{t} \), we substitute the known values: 195 for \(d\) and 3 for \(t\), giving us \[ r = \frac{195}{3} = 65 \] Thus, the unknown variable \(r\) is resolved as 65. Isolating the unknown variable is crucial, as it clarifies the solution path and simplifies the calculations needed.