Problem 23 - Brief Applied Calculus

Question

For each equation, find the slope \(m\) and \(y\) -intercept \((0, b)\) (when they exist) and draw the graph. \(2 x-3 y=12\)

Step-by-Step Solution

Answer

The slope is \(m = \frac{2}{3}\) and the y-intercept is \((0, -4)\).

Step 1: Convert to slope-intercept form

To find the slope and the y-intercept, the equation needs to be in the slope-intercept form, which is \(y = mx + b\). Starting with the given equation \(2x - 3y = 12\), we need to solve for \(y\). Begin by isolating the \(y\)-term: Subtract \(2x\) from both sides: \[-3y = -2x + 12\] Next, divide every term by \(-3\) to solve for \(y\): \[y = \frac{2}{3}x - 4\].Now, the equation is in slope-intercept form.

Step 2: Identify slope and y-intercept

From the equation \(y = \frac{2}{3}x - 4\), we can directly identify the slope \(m\) and the y-intercept \(b\). Here:- The slope \(m = \frac{2}{3}\).- The y-intercept is where the line crosses the y-axis, so the y-intercept is \(b = -4\). Thus, the point is \((0, -4)\).

Step 3: Draw the graph

To draw the graph, use the slope and y-intercept:1. Start by plotting the y-intercept \((0, -4)\) on the coordinate plane.2. Use the slope \(m = \frac{2}{3}\) to find another point. From \((0, -4)\), move up 2 units and right 3 units to reach the point \((3, -2)\).3. Draw a straight line through the points \((0, -4)\) and \((3, -2)\) to graph the equation.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a line so that it is easy to graph. It's expressed as \( y = mx + b \), where:

• \( m \) represents the slope of the line.
• \( b \) is the y-intercept, which is where the line crosses the y-axis.

To convert an equation into this form, you simply solve for \( y \). For example, if you start with the equation \( 2x - 3y = 12 \), the goal is to rearrange it so that it fits the \( y = mx + b \) format. By isolating \( y \), you change it step by step until you end up with \( y = \frac{2}{3}x - 4 \), which clearly shows the slope and the y-intercept. This form makes it very easy to both identify key characteristics of the line and graph it quickly.

Finding the Slope

The slope is a measure of a line's steepness and direction on a graph. In the slope-intercept form \( y = mx + b \), the slope is the coefficient \( m \) of \( x \). A positive \( m \) means the line rises as it goes from left to right, while a negative \( m \) indicates the line falls.In our example, the slope of the line is \( \frac{2}{3} \). This means for every 3 units you move to the right on the x-axis, the line moves up 2 units on the y-axis. This rise over run is a simple way to understand and visualize the slope on a graph. Knowing the slope is essential for graphing and understanding the behavior of linear equations.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis on a graph. It's a crucial reference point when graphing linear equations. In slope-intercept form \( y = mx + b \), \( b \) stands for the y-intercept. This point is expressed as \((0, b)\) since the x-coordinate is always 0.For the equation \( y = \frac{2}{3}x - 4 \), the y-intercept is \( -4 \). You plot this point directly on the y-axis at \((0, -4)\). This point serves as a starting location for drawing the graph line. By plotting this point and using the slope to find another point, you can accurately draw the entire line on a graph. Understanding the y-intercept helps ensure your graph is correctly positioned on the coordinate plane.