Problem 62 - Calculus

Question

Explain what is wrong with the statement. The equation \(y=2 x+1\) indicates that \(y\) is directly proportional to \(x\) with a constant of proportionality 2

Step-by-Step Solution

Answer

The equation has a y-intercept (1), so it's not direct proportionality.

Step 1: Define Direct Proportionality

In mathematics, two variables are said to be directly proportional if the relationship between them can be expressed as \( y = kx \) where \( k \) is a constant. This means that when one variable changes, the other changes in a proportional manner, and the line passes through the origin \((0,0)\).

Step 2: Analyze the Given Equation

The given equation is \( y = 2x + 1 \). This can be rearranged to the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here, \( m = 2 \) and \( b = 1 \).

Step 3: Determine Y-Intercept Impact

The presence of the y-intercept \( b = 1 \) means that the line does not pass through the origin. For a relationship to be directly proportional, \( b \) must equal \( 0 \).

Step 4: Identify the Error

Since the equation \( y = 2x + 1 \) includes a y-intercept other than zero, it does not represent direct proportionality. The definition of direct proportionality requires no additional constant \( b \) in the equation other than \( k \).

Key Concepts

Linear Equations

Linear equations are foundational in both mathematics and science. They can be easily recognized by their structure, typically expressed in the form of \( y = mx + b \). Here, \( m \) represents the slope, which indicates the steepness of the line. Meanwhile, \( b \), known as the y-intercept, represents the point where the line crosses the y-axis. This form is called the "slope-intercept form". Linear equations grasp a broad range of relationships by showing how a change in one variable leads to a change in another.Understanding the slope is essential. It reflects the direction and steepness of a line. A positive slope means the line is going upward as it moves from left to right, while a negative slope shows the line going downward. Slopes also show how quickly one variable changes in relation to another. However, every linear equation will graph as a straight line, making them easy to visualize.Every linear equation can tell a simple story of consistency and predictability. This is because they will always produce straight lines without curves. Understanding these equations is crucial in finding and interpreting relationships in various real-life situations.

Y-Intercept

The y-intercept is a critical component of a linear equation, particularly in the slope-intercept form \( y = mx + b \). It represents the point where the line intersects the y-axis.When you look at a graph, the y-intercept \( b \) gives you a clear starting point for plotting your line. This point occurs when \( x = 0 \) in the equation. For our example of \( y = 2x + 1 \), the y-intercept is 1. This means if you were to draw the line on a graph, it would cross the y-axis at the point (0,1).To further illustrate, consider the implication of a y-intercept in real-life scenarios. If the equation represents a financial model, the y-intercept might indicate a starting cost. Essentially, it's the point before any changes (in terms of x) have been made. Recognizing and accurately calculating the y-intercept is key to understanding where your linear function starts on the graph.

Proportional Relationships

Proportional relationships are unique and simple yet offer a wide range of applications. They occur when two quantities increase or decrease at the same rate. Mathematically, this is expressed as \( y = kx \), where \( k \) is a constant known as the constant of proportionality.A key feature of proportional relationships is that they always graph as a line passing through the origin \((0,0)\). This means that there is no added value, such as a y-intercept \( b \), in the equation. The graph of direct proportionality is always a straight line that extends through both quadrants depending on the value of \( k \).Proportional relationships are straightforward yet powerful. In practice, they make it easier to calculate rates and ratios, like speed (distance/time) or density (mass/volume). In our case, the equation \( y=2x+1 \) is mistakenly identified as directly proportional, as the presence of the y-intercept \( b \) breaks its qualification as a proportional relationship.