91Ó°ÊÓ

In direct variation problems, the proportionality constant, often denoted as \(k\), is a key value that links the dependent variable to the independent variable. Imagine it as a multiplier that adjusts the independent variable to give you the dependent variable.
To find \(k\), you substitute known values of both variables into the direct variation equation. Here's how it works in the given exercise:
The office requires 16 cartridges when there are 2800 students. This gives us the equation \[16 = k \cdot 2800\]\br> Solving for \(k\), we divide both sides by 2800:
\[k = \frac{16}{2800} = \frac{2}{350} = \frac{1}{175}\]\br> Now, we have the proportionality constant, \(k = \frac{1}{175}\).

An equation of variation shows how one variable changes directly with another. This means that if one variable increases, so does the other, and if one decreases, the other does, too. It's written in the form \I = k \cdot s\, where \(I\) is the dependent variable, \(s\) is the independent variable, and \(k\) is the proportionality constant.
Using the \(k\) we found earlier, we can write the specific equation of variation for our exercise:
\[I = \frac{1}{175} \cdot s\]\br> This means that for every additional student enrolled, the office requires \frac{1}{175}\ more inkjet cartridges. You can plug any value of \(s\) (number of students) into this equation to find the corresponding \(I\) (number of cartridges).

The substitution method involves replacing variables in an equation with known values to find a missing variable. We often use this method for solving equations, especially in direct variation problems.
In the exercise, after finding the direct variation equation as \(I = \frac{1}{175} \cdot s\), we need to determine the number of cartridges for 3100 students.
We substitute \(s = 3100\) into the equation:
\[I = \frac{1}{175} \cdot 3100\]
Next, we calculate the value:
\[I = \frac{3100}{175} = 17.7143\]
This rounds to approximately 18 cartridges.
Substitution is a straightforward and powerful method to solve for unknowns using known values, helping us find the outcomes quickly and accurately in direct variation scenarios.