Question

a. By using x and y, state the general equation for quantities that are directly proportional. b. For two directly proportional quantities, what happens to one variable when the other variable increases?

Short answer

a) General equation: \( y = kx \). b) When one variable increases, the other variable also increases.

Step-by-step solution

Understanding Direct Proportion

When two quantities are directly proportional, it means that as one quantity increases, the other quantity increases at a constant rate. Similarly, if one decreases, the other also decreases at the same rate.

General Equation for Direct Proportion

The general equation for two directly proportional quantities is: \[ y = kx \] where \(y\) and \(x\) are the variables, and \(k\) is the constant of proportionality.

Relationship Between Variables

When two variables are directly proportional, if one variable \((x)\) increases, the other variable \((y)\) also increases. Conversely, if \(x\) decreases, \(y\) also decreases.

Key concepts

General Equation

In mathematics, when we talk about two quantities being directly proportional, we mean that they change together at a consistent rate. This consistent relationship between the quantities can be described using a general equation. To express this relationship, we use the equation: \[ y = kx \] Where:

• \(y\): represents one variable
• \(x\): represents the other variable
• \(k\): is the constant of proportionality, a fixed number that doesn’t change

This equation tells us that if you know one of the variables (either \(x\) or \(y\)), and you know \(k\), you can always find out the other variable.

Constant Rate

When two quantities are directly proportional, they change at a constant rate relative to each other. This means that if one quantity doubles, the other quantity also doubles. If one quantity is halved, the other is also halved. This is reflected in our general equation \(y = kx\). Here’s how it breaks down:

• If \(x\) increases, \(y\) increases by the multiplication factor of \(k\)
• If \(x\) decreases, \(y\) decreases in the same proportion

Think of it like this: if you walk at a constant speed, the distance you cover increases steadily over time. If you speed up or slow down, the distance still increases or decreases consistently with your speed.

Constant of Proportionality

The constant of proportionality, symbolized as \(k\), is a key factor in direct proportionality. It represents the consistent rate at which the two variables change relative to each other. In the equation \(y = kx\), \(k\) is what links \(x\) and \(y\) together in a fixed ratio. Here’s what you should know about \(k\):

• \(k\) determines how much one variable changes in response to the other
• \(k\) is always a non-zero constant
• The value of \(k\) can be any real number, positive or negative

For example, if \(k\) is 3, it means that for every 1 unit increase in \(x\), \(y\) increases by 3 units. If \(k\) is -2, it means that for every 1 unit increase in \(x\), \(y\) decreases by 2 units. Understanding \(k\) is crucial because it helps you predict the behavior of the variables in a directly proportional relationship.