Problem 30 - Brief Applied Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the standard form

The given equation is \( \frac{x}{2} + \frac{y}{3} = 1 \). This is in the form \( Ax + By = C \). Here, \( A = \frac{1}{2} \), \( B = \frac{1}{3} \), and \( C = 1 \).

Step 2: Convert to slope-intercept form

To find the slope and y-intercept, convert the equation to the slope-intercept form \( y = mx + b \). Start by isolating \( y \):\[ \frac{y}{3} = 1 - \frac{x}{2}.\] Multiply everything by 3 to clear the fraction:\[ y = 3 \left(1 - \frac{x}{2}\right). \] Simplify it to \[ y = -\frac{3}{2}x + 3. \]

Step 3: Identify slope and y-intercept

Now that we have the equation in the slope-intercept form \( y = mx + b \), we can easily identify the slope \( m \) and y-intercept \( b \). From \( y = -\frac{3}{2}x + 3 \), the slope \( m \) is \(-\frac{3}{2}\) and the y-intercept \( b \) is 3.

Step 4: Graph the equation

Start by plotting the y-intercept \( (0, 3) \) on the graph. Then, use the slope \( -\frac{3}{2} \) to find another point. From \( (0, 3) \), move down 3 units and to the right 2 units, which lands you at the point \( (2, 0) \). Draw a line through the points \( (0, 3) \) and \( (2, 0) \). This line is the graph of the equation.

Key Concepts

Graphing Linear Equations

Graphing linear equations is a visual method of representing solutions to equations. When dealing with a linear equation like \( \frac{x}{2} + \frac{y}{3} = 1 \), you first need to understand that it represents a straight line on a coordinate plane. Each point on this line is a solution to the equation.
To graph a linear equation, follow these steps:

• First, convert the equation into a form that is easier to work with, typically the slope-intercept form, which is \( y = mx + b \).
• The next step is to determine at least two points that the line passes through. This can be done by finding the y-intercept and using the slope to locate another point.
• Once you have these two points, you draw a line through them, which will be the graph of the equation.

By following these steps, you will have a clear visual representation of the equation. Graphing is essential because it allows you to see relationships and trends in the data represented by the equation.

Y-Intercept

The y-intercept of a linear equation is the point where the line crosses the y-axis on a graph. In the slope-intercept form \( y = mx + b \), the y-intercept is represented by \( b \). This is because, at the y-intercept, the value of \( x \) is always 0.
For the equation \( y = -\frac{3}{2}x + 3 \), the y-intercept \( b \) is 3. This means that when the line crosses the y-axis, it does so at the point \( (0, 3) \).
Having a clear understanding of the y-intercept is crucial because it gives you a starting point on a graph. From this point, you can easily apply the slope to determine other points on the line. Always remember:

• The y-intercept tells you the initial value of \( y \) when \( x = 0 \).
• It acts as a key point in plotting linear equations.

Identifying the y-intercept helps in setting the foundation for the entire graph and simplifies finding other points along the line.

Slope Calculation

Slope is a measure of the steepness of a line, depicted as \( m \) in the slope-intercept form \( y = mx + b \). It tells us how much \( y \) changes for a unit change in \( x \). A positive slope means the line rises as \( x \) increases, while a negative slope means it falls.
In our equation, \( y = -\frac{3}{2}x + 3 \), the slope \( m \) is \(-\frac{3}{2}\). Here's how to interpret it:

• The negative sign indicates that for every increase of 2 units in \( x \), \( y \) decreases by 3 units.
• This forms a downward slope from left to right on a graph.

To calculate and use the slope effectively:

• Start from a known point like the y-intercept \((0, 3)\).
• Apply the slope: Move down 3 units and right 2 units to find another point \((2, 0)\).

Understanding slope calculation is vital as it directly affects how you draw and interpret the graph of the equation. The slope provides insight into the relationship between the x and y variables.