Problem 49 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of the given equation. Label the intercepts. $$y=\frac{x-3}{2}$$

Step-by-Step Solution

Answer

The intercepts are \( (0, -\frac{3}{2}) \) and \( (3, 0) \).

Step 1: Identify the type of equation

The given equation is a linear equation in the form of \( y = \frac{x-3}{2} \). This can be rewritten to better recognize it as a linear equation: \( y = \frac{1}{2}x - \frac{3}{2} \)

Step 2: Determine the slope and y-intercept

The slope-intercept form of a linear equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In the equation \( y = \frac{1}{2}x - \frac{3}{2} \), the slope \( m \) is \( \frac{1}{2} \) and the y-intercept \( b \) is \( -\frac{3}{2} \).

Step 3: Find the y-intercept

To find the y-intercept, set \( x = 0 \) in the equation \( y = \frac{1}{2}x - \frac{3}{2} \):\[ y = \frac{1}{2}(0) - \frac{3}{2} \]\[ y = -\frac{3}{2} \].The y-intercept is \( (0, -\frac{3}{2}) \).

Step 4: Find the x-intercept

To find the x-intercept, set \( y = 0 \) in the equation \( y = \frac{1}{2}x - \frac{3}{2} \): \[ 0 = \frac{1}{2}x - \frac{3}{2} \]Solve for \( x \):\[ \frac{3}{2} = \frac{1}{2}x \]\[ x = 3 \].The x-intercept is \( (3, 0) \).

Step 5: Plot the intercepts

On a coordinate plane, plot the y-intercept \( (0, -\frac{3}{2}) \) and the x-intercept \( (3, 0) \).

Step 6: Draw the line

Draw a straight line through the two intercept points. This line represents the graph of the equation \( y = \frac{1}{2}x - \frac{3}{2} \).

Key Concepts

Graphing Linear Equations

Graphing linear equations requires understanding the equation's form and how to interpret it visually. Linear equations create straight lines on a graph.
An equation like \( y = \frac{1}{2}x - \frac{3}{2} \) can be rewritten or simplified to help identify key characteristics. In this case, it's already in slope-intercept form, \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

To graph a linear equation:

• Start by identifying the y-intercept (where the line crosses the y-axis).
• Next, use the slope to determine the direction and steepness of the line.
• You can plot additional points if needed to ensure accuracy, but finding the intercepts is often enough.

Understanding these steps is crucial for visualizing and graphing linear equations accurately.

Slope and Y-Intercept

The slope and y-intercept are critical components of linear equations.
The slope (\( m \)) indicates the rise over run, or how steep the line is. In \( y = \frac{1}{2}x - \frac{3}{2} \), the slope \( \frac{1}{2} \) shows that for each unit increase in \( x \), \( y \) increases by \( \frac{1}{2} \). This results in a gently rising line.

The y-intercept (\( b \)) is where the line crosses the y-axis. For our example, \( b \) is \( -\frac{3}{2} \), meaning the line crosses the y-axis at \( y = -\frac{3}{2} \).

By understanding the slope and y-intercept, you can quickly sketch the line:

• The y-intercept point is \( (0, -\frac{3}{2}) \).
• Use the slope to find another point, such as moving one unit right and \( \frac{1}{2} \) units up from the y-intercept.
• Connect these points to draw the line.

This method ensures that your line accurately represents the equation.

Finding Intercepts

To graph a linear equation, finding the intercepts is a fundamental step. The intercepts are where the line crosses the axes.

The y-intercept is found by setting \( x = 0 \) in the equation:
In \( y = \frac{1}{2}x - \frac{3}{2} \):

• Set \( x = 0 \)
• Calculate \( y = \frac{1}{2}(0) - \frac{3}{2} = -\frac{3}{2} \)

This result gives the y-intercept at point \( (0, -\frac{3}{2}) \).

To find the x-intercept, set \( y = 0 \):
In \( y = \frac{1}{2}x - \frac{3}{2} \):

• Set \( y = 0 \)
• Solve: \( 0 = \frac{1}{2}x - \frac{3}{2} \)
• Rearrange to find \( x = 3 \)

This result gives the x-intercept at point \( (3, 0) \).

Intercepts are crucial because they provide specific points to plot on the graph, making it easier to draw the accurate line of the equation.