Problem 49 - Precalculus Mathematics for Calculus

Question

Find the slope and \(y\) -intercept of the line and draw its graph. $$\frac{1}{2} x-\frac{1}{3} y+1=0$$

Step-by-Step Solution

Answer

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

Step 1: Rewrite the Equation in Slope-Intercept Form

The slope-intercept form of a line is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Let's manipulate the given equation \(\frac{1}{2}x - \frac{1}{3}y + 1 = 0\) to this form.\\[ -\frac{1}{3}y = -\frac{1}{2}x - 1 \]. Multiply the whole equation by \(-3\) to isolate \(y\):\\[ y = \frac{3}{2}x + 3 \].

Step 2: Identify the Slope and Y-Intercept

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

Step 3: Plot the Y-Intercept

Start plotting the line by marking the y-intercept on the graph. The y-intercept, \(b = 3\), means the line crosses the y-axis at the point \((0, 3)\).

Step 4: Use the Slope to Find Another Point

With a slope of \(\frac{3}{2}\), for every 2 units you move to the right on the x-axis, you move 3 units up on the y-axis. From the y-intercept \((0, 3)\), moving 2 units to the right gives \((2, 6)\).

Step 5: Draw the Line

Plot the points \((0, 3)\) and \((2, 6)\) on the graph, and draw a line through these points extending in both directions. This represents the graph of the equation \(y = \frac{3}{2}x + 3\).

Key Concepts

Slope

In a linear equation expressed in the slope-intercept form, the slope is a crucial element that tells you how steep the line is. It's represented by the letter \(m\). The slope is calculated based on how quickly the \(y\) value changes to the \(x\) value. In the coordinate system, we say that the slope is the "rise over run," or the change in \(y\) over the change in \(x\).
For example, if an equation is written as \(y = \frac{3}{2}x + 3\), the slope, \(m\), is \(\frac{3}{2}\). This means:

• For every 2 units you move horizontally (to the right), the line rises vertically (up) by 3 units.
• If you move horizontally 2 units to the left, you go down 3 units vertically.

Understanding this makes it easier to draw the line accurately on a graph. Remember, slopes can be positive, negative, zero, or undefined, depending on the direction and steepness of the line:

Y-Intercept

The y-intercept of a line is where it crosses the y-axis. It's like the starting point of the line if you were walking from left to right on the graph. You'll find it by looking at the \(b\) value in the slope-intercept form \(y = mx + b\).
For our example, \(y = \frac{3}{2}x + 3\), the y-intercept \(b\) is 3. So, the line crosses the y-axis at \( (0, 3) \). This tells you that no matter what value of \(x\), if \(x=0\), \(y\) will always be 3.
The y-intercept is very important, as it helps you start plotting the line. Once you know where the line starts on the vertical axis, using the slope, you can plot any additional points needed to draw the line accurately.

Graphing Linear Equations

Graphing linear equations involves a series of straightforward steps once you understand the slope and the y-intercept. Let's walk through these steps using the equation \(y = \frac{3}{2}x + 3\).
First, start by plotting the y-intercept on the graph. In this case, the y-intercept is 3, so the point is \((0, 3)\). This is where your line will intersect the y-axis. Next, utilize the slope, which is \(\frac{3}{2}\), to determine the direction and angle of your line.

• From the point \((0, 3)\), move 2 units to the right along the \(x\)-axis.
• From there, move up 3 units parallel to the \(y\)-axis.
• Plot that second point, which will be at \((2, 6)\).

Now that you have two points, you can draw a straight line through them. Extend this line in both directions to complete the graph of the equation. This process allows you to visualize the relationship between \(x\) and \(y\), and with practice, it becomes an intuitive way to graph linear equations.