Problem 28 - Applied Calculus

Question

For each equation, find the slope \(m\) and \(y\) -intercept \((0, b)\) (when they exist) and draw the graph. $$ y=\frac{2}{3}(x-3) $$

Step-by-Step Solution

Answer

The slope is \(\frac{2}{3}\) and the y-intercept is \((0, -2)\).

Step 1: Expand the Equation

Start with the given equation: \( y = \frac{2}{3}(x - 3) \).To simplify it, first distribute \( \frac{2}{3} \) across the terms inside the parentheses:\[y = \frac{2}{3}x - \frac{2}{3} \times 3\]This simplifies to:\[y = \frac{2}{3}x - 2\]

Step 2: Identify the Slope

In the linear equation \(y = mx + b\), \(m\) represents the slope. From the simplified equation \(y = \frac{2}{3}x - 2\), we can see that the slope \(m\) is \(\frac{2}{3}\).

Step 3: Identify the Y-Intercept

Continuing with the equation \(y = \frac{2}{3}x - 2\), the term \(-2\) represents the \(y\)-intercept, \(b\). This means the \(y\)-intercept is \((0, -2)\).

Step 4: Sketch the Graph

To graph the equation, plot the \(y\)-intercept \((0, -2)\) on the graph.Then, use the slope \(\frac{2}{3}\) which means: from \((0, -2)\), move up 2 units and right 3 units to plot another point.Draw a straight line through these points to complete the graph.

Key Concepts

Understanding Slope

The slope of a line is a fundamental concept in linear equations. It represents the steepness or the angle of the line, showing how much the line rises or falls as you move from left to right. In mathematical terms, the slope is defined as the change in the y-coordinate divided by the change in the x-coordinate. This is often described using "rise over run," indicating how many units the line goes up (or down) for each unit it goes right. For the equation \( y = \frac{2}{3}x - 2 \), the slope is \( \frac{2}{3} \). Here's what this means:

• The rise is 2 units, meaning it goes up 2 units vertically.
• The run is 3 units, meaning for those 2 units vertically, it moves 3 units horizontally to the right.

So, the line ascends gently as you move along the x-axis. Slopes can be positive, negative, zero, or undefined. A positive slope like our example means the line goes upwards, while a negative slope means it descends. A zero slope implies the line is horizontal, and an undefined slope means the line is vertical.

Identifying the Y-Intercept

The y-intercept is a crucial point where the line crosses the y-axis. It's represented in an equation by the term \( b \) in the standard form \( y = mx + b \). This value tells us exactly where the line will intersect the y-axis, which is the point when \( x = 0 \). In the equation \( y = \frac{2}{3}x - 2 \), the y-intercept \( b \) is \(-2\). This means the line crosses the y-axis at the point \( (0, -2) \).To visualize this:

• Imagine starting at the origin on a graph where both x and y are zero.
• Move directly down 2 units along the y-axis, since our intercept is \(-2\).

The y-intercept provides a starting point for graphing the line and is essential for understanding the line's position relative to both the x and y-axes.

Graphing Linear Equations

Graphing linear equations involves plotting points and drawing a straight line through them. It's a visual way to represent the equation and see how the values change. To correctly graph the equation \( y = \frac{2}{3}x - 2 \), follow these steps:First, locate the y-intercept. We've found it already: \( (0, -2) \). Plot this point first on the graph.Next, use the slope to find another point. For our slope \( \frac{2}{3} \):

• Start at the y-intercept point \( (0, -2) \).
• Since the slope indicates "rise over run," move up 2 units (rise) and 3 units to the right (run).
• Mark this new point, \( (3, 0) \), on the graph.

Finally, draw a line through these two points. Extend it both ways to cover the entire graph. The points you've plotted and the line you draw represent all the solutions to the equation, showing all the combinations of \( x \) and \( y \) that satisfy it.Graphing helps to see the equation's behavior, making it easier to understand different relationships between variables.