Problem 24 - Brief Applied Calculus

Question

For each equation, find the slope \(m\) and \(y\) -intercept \((0, b)\) (when they exist) and draw the graph. \(3 x+2 y=18\)

Step-by-Step Solution

Answer

Slope: \(-\frac{3}{2}\), Y-intercept: \((0,9)\).

Step 1: Identify the Equation Type

We have the equation \(3x + 2y = 18\). To find the slope \(m\) and the \(y\)-intercept \(b\), we need to convert this equation into the slope-intercept form \(y = mx + b\).

Step 2: Convert to Slope-Intercept Form

To do this, solve for \(y\). Start by isolating the \(y\) term. Move \(3x\) to the other side of the equation:\[2y = -3x + 18\]

Step 3: Solve for y

Divide each term by 2 to isolate \(y\):\[y = -\frac{3}{2}x + 9\]

Step 4: Determine the Slope and Y-Intercept

The equation is now in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept. Here, \(m = -\frac{3}{2}\) and the \(y\)-intercept is \(b = 9\). The \(y\)-intercept point on the graph is \((0, 9)\).

Step 5: Graph the Equation

Start by plotting the \(y\)-intercept \((0, 9)\) on the graph. Next, use the slope \(-\frac{3}{2}\) (which means down 3 and right 2) to find another point. From \((0, 9)\), move down 3 and right 2 to point \((2, 6)\). Draw a line through these two points to graph the equation.

Key Concepts

Graphing Linear Equations

When graphing linear equations, it's all about understanding the relationship between variables and how they depict a line on a graph. A linear equation is typically written in the slope-intercept form, which is expressed as \( y = mx + b \). Here, \( m \) represents the slope, which indicates the steepness and direction of the line, and \( b \) is the \( y \)-intercept, which is where the line crosses the \( y \)-axis. By converting any given linear equation to this form, you make it easier to graph.

Once you have the equation in the slope-intercept form:

• Identify the \( y \)-intercept and plot it on the graph.
• Use the slope to determine the direction and steepness of the line, allowing you to find another point.
• Draw a straight line through these points to complete the graphing process.

This method gives a clear visual representation of the equation, making it easier to understand the underlying relationship between the variables.

Y-Intercept

The \( y \)-intercept is a critical component when graphing linear equations. It is denoted by \( b \) in the slope-intercept form, \( y = mx + b \). The \( y \)-intercept is the point where the line crosses the \( y \)-axis, specifically at the coordinates \( (0, b) \). This means that at this point, the value of \( x \) is zero.

To identify the \( y \)-intercept:

• Convert the linear equation into the slope-intercept form.
• Look for the constant term \( b \) which represents the \( y \)-intercept.
• Mark this point on the graph as it helps in setting up the line direction.

Understanding the \( y \)-intercept is essential because it provides a starting point for graphing the line. It's the first step in plotting a linear equation and gives you a fixed reference to apply the slope.

Slope Calculation

The slope of a linear equation provides valuable information about the line's angle and direction. In the slope-intercept form \( y = mx + b \), the term \( m \) represents the slope. It tells you how steep the line is and whether it ascends or descends. A positive slope indicates an upward trend, while a negative slope indicates a downward trend.

For calculating and using the slope:

• Convert the equation to \( y = mx + b \) form to identify \( m \).
• Understand that \( m = \frac{\text{rise}}{\text{run}} \), where "rise" is the change in \( y \) and "run" is the change in \( x \).
• If the slope is \( -\frac{3}{2} \), for example, it means for every 3 units you move downwards, you go 2 units to the right.

Using the slope helps in plotting another point starting from the \( y \)-intercept. It's crucial for accurately drawing the line on a graph by dictating how you move from one point to the next.