Problem 1 - Precalculus: Mathematics for Calculus

Question

If the quantities \(x\) and \(y\) are related by the equation \(y=3 x\),then we say that \(y\) is _______________ _______________to \(x\) and the constant of _______________ is 3

Step-by-Step Solution

Answer

y is directly proportional to x and the constant of proportionality is 3.

Step 1: Identify the Relationship Type

We are given the equation \( y = 3x \). This is a linear relationship between \( y \) and \( x \). In such relationships, if \( y \) is expressed as a constant multiplied by \( x \), it indicates that \( y \) is directly proportional to \( x \).

Step 2: Determine the Constant of Proportionality

In a direct proportionality relationship, \( y = kx \), where \( k \) is the constant of proportionality. Here, in the equation \( y = 3x \), the coefficient of \( x \) is 3. Therefore, the constant of proportionality is 3.

Key Concepts

Linear Relationship

Understanding the concept of a linear relationship is essential when studying equations like \( y = 3x \). A linear relationship involves a direct connection between two variables, such that when one variable changes, the other changes at a constant rate. This means, whenever you plot this relationship on a graph, it will always form a straight line. The term "linear" itself signifies that the relationship between the variables can be represented by a line.

Linear relationships are foundational in algebra and reflect a simple yet powerful connection between quantities. In our example, the equation \( y = 3x \) shows that for every increase in \( x \), \( y \) increases three times as much. This consistent change reflects the essence of a linear relationship.

• A graph of such a relationship will be a straight line.
• The slope of the line is constant, reflecting the consistent rate of change.
• The formula for a linear relationship is generally expressed in the form \( y = kx \).

This simplicity allows for easy prediction and calculation of one quantity based on the knowledge of the other.

Constant of Proportionality

The constant of proportionality is a crucial element in understanding direct relationships between two variables. It is the factor that relates two quantities in a proportional way. In the equation \( y = 3x \), the number 3 acts as the constant of proportionality. This signifies that \( y \) changes three times faster than \( x \) does – a very specific and defined relationship.

When we talk about direct proportionality, we mean that two variables increase or decrease at the same rate, relative to each other. This is quantified by the constant of proportionality.

• The constant is always the coefficient of \( x \) in the equation.
• It signifies the slope of the line in a graph of the relationship.
• Understanding this constant helps in predicting values effectively.

The constant of proportionality simplifies the relationship between variables, making it easier to manage and grasp the changes that occur between them.

Equation of a Line

The equation of a line is fundamental in representing linear relationships. In our example, the equation \( y = 3x \) neatly captures how the two quantities, \( y \) and \( x \), relate to each other in a precise, mathematical form.

This equation can be termed as the slope-intercept form of a linear equation, typically expressed as \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept. In \( y = 3x \), the y-intercept \( b \) is 0, and the slope \( m \) is 3, which tells us that the line crosses the origin and rises three units for every one unit along the x-axis.

• The slope is essential for determining how steep the line is.
• The y-intercept indicates where the line crosses the y-axis.
• This form of equation helps easily sketch graphs and understand relationships.

The equation of a line allows easy visualization and better understanding of linear relationships, making it easier to find solutions and predict outcomes.