Problem 31 - Applied Calculus

Question

For each equation, find the slope \(m\) and \(y\) -intercept \((0, b)\) (when they exist) and draw the graph. $$ \frac{2 x}{3}-y=1 $$

Step-by-Step Solution

Answer

Slope \( m = \frac{2}{3} \), y-intercept \((0, -1)\).

Step 1: Rewrite in Slope-Intercept Form

Start by rewriting the given equation \( \frac{2x}{3} - y = 1 \) into the standard slope-intercept form \( y = mx + b \). To do this, solve for \( y \):\[ \frac{2x}{3} - y = 1 \]. Add \( y \) to both sides to get: \[ \frac{2x}{3} = y + 1 \]. Next, subtract 1 from both sides: \[ \frac{2x}{3} - 1 = y \]. Finally, rewrite it in the form \( y = mx + b \): \( y = \frac{2}{3}x - 1 \).

Step 2: Identify the Slope and Y-Intercept

The equation \( y = \frac{2}{3}x - 1 \) is now in the standard form \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept. Thus, the slope \( m \) is \( \frac{2}{3} \), and the \( y \)-intercept \( b \) is \( -1 \). Therefore, the point for the \( y \)-intercept is \((0, -1)\).

Step 3: Draw the Graph

To draw the graph of the line, start by plotting the \( y \)-intercept, which is the point \((0, -1)\) on the y-axis. Next, use the slope \( \frac{2}{3} \) to plot the next point. The slope indicates that for every 3 units moved to the right (in the positive \( x \)-direction), the graph rises by 2 units (in the positive \( y \)-direction). From the point \((0, -1)\), move 3 units to the right to \((3, -1)\) and then 2 units up to \((3, 1)\). Draw a straight line through these points to get the graph of the line.

Key Concepts

Graphing Linear Equations

When graphing linear equations, the goal is to draw a line that represents all the solutions of the equation. A linear equation is an equation that forms a straight line when graphed, and it can be represented in various forms, with the slope-intercept form being the most common. To begin graphing, start by identifying at least two points on the line, as a straight line is defined by just two points.

Here’s how:

• Find the y-intercept, the point where the line crosses the y-axis.
• Use the slope, which is the ratio of the change in y to the change in x, to determine another point.

Once you have these points, draw a straight line through them, extending in both directions. Remember, the line continues infinitely, so be sure to stretch it across the graph. This visual representation helps understand the relationship described by the equation.

Finding Slope and Y-Intercept

To master graphing, it's crucial to understand how to find the slope and y-intercept of a linear equation. The slope, represented as \( m \), indicates how steep the line is. The y-intercept, denoted by \( b \), is where the line crosses the y-axis.

Follow these steps:

• The slope \( m \) can be understood as "rise over run." It tells you how many units the line goes up or down for every unit it goes horizontally to the right.
• The y-intercept \( b \) is simply the y-coordinate of the point where the line meets the y-axis.

In the equation \( y = \frac{2}{3}x - 1 \), the slope \( m \) is \( \frac{2}{3} \), meaning the line goes up 2 units for every 3 units you move to the right. The y-intercept \( b \) is \(-1\), telling us that the line crosses the y-axis at the point \((0, -1)\). This understanding helps predict the behavior of the line.

Rewriting Equations

Rewriting equations into different forms can make them easier to work with, especially when graphing. The slope-intercept form of a linear equation is particularly useful because it directly shows the slope \( m \) and y-intercept \( b \).

Here’s how you can transform an equation into this form:

• Start with the original equation, such as \( \frac{2x}{3} - y = 1 \).
• Rearrange it to solve for \( y \). This involves moving \( y \) to one side and the other terms to the opposite side.
• In our example, by adding \( y \) and then subtracting 1, you get \( y = \frac{2}{3}x - 1 \).

This form quickly reveals the slope and y-intercept, making graphing straightforward. Rewriting equations can simplify problem-solving and provide greater insight into the relationships expressed by the equation.