Problem 42 - College Algebra with Corequisite Support

Question

For each of the following exercises, find and plot the \(x\) - and \(y\) -intercepts, and graph the straight line based on those two points. $$ y=\frac{x-3}{2} $$

Step-by-Step Solution

Answer

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

Step 1: Find the y-intercept

The y-intercept occurs when the x-value is 0. Substitute \( x = 0 \) into the equation \( y = \frac{x-3}{2} \). \[y = \frac{0-3}{2} = \frac{-3}{2}\]Thus, the y-intercept is at the point \( (0, -\frac{3}{2}) \).

Step 2: Find the x-intercept

The x-intercept occurs when the y-value is 0. Substitute \( y = 0 \) into the equation \( y = \frac{x-3}{2} \) and solve for \( x \).\[0 = \frac{x-3}{2}\]Multiply both sides by 2:\[0 = x - 3\]Add 3 to both sides:\[x = 3\]Thus, the x-intercept is at the point \( (3, 0) \).

Step 3: Plot the intercepts

To graph the line, plot the points of the intercepts found in Step 1 and Step 2: 1. Plot the point \( (0, -\frac{3}{2}) \) for the y-intercept.2. Plot the point \( (3, 0) \) for the x-intercept.

Step 4: Sketch the line

Draw a straight line through the intercept points plotted in Step 3. This line represents the equation \( y = \frac{x-3}{2} \), and extends infinitely in both directions.

Key Concepts

Understanding the x-intercepts

The x-intercept of a linear equation is where the graph crosses the x-axis. This is the point where the y-value is zero because any point along the x-axis has a y-coordinate of zero.
To find the x-intercept of the equation \( y = \frac{x-3}{2} \), we set \( y = 0 \) and solve for \( x \). The equation becomes:

• \(0 = \frac{x-3}{2}\)
• Multiply both sides by 2 to eliminate the fraction: \( 0 = x - 3 \)
• Add 3 to both sides to solve for \( x \): \( x = 3 \)

Thus, the x-intercept is at the point \((3, 0)\). Knowing how to find the x-intercept is a key skill in graphing linear equations, as it helps to anchor one end of the line.

Exploring the y-intercepts

The y-intercept of a linear equation is where the graph crosses the y-axis. At this point, the x-coordinate is always zero because any point along the y-axis has an x-coordinate of zero.
To find the y-intercept for the equation \( y = \frac{x-3}{2} \), we substitute \( x = 0 \) into the equation:

• Substitute and calculate: \( y = \frac{0-3}{2} \)
• Simplify the calculation: \( y = \frac{-3}{2} \)

Therefore, the y-intercept is at the point \((0, -\frac{3}{2})\). Understanding y-intercepts helps you graph equations quickly, as it gives you a starting point from the vertical axis.

Graphing linear equations accurately

Graphing a linear equation starts with plotting the intercepts you found.
Place your first point at the y-intercept \((0, -\frac{3}{2})\) and your second point at the x-intercept \((3, 0)\).
With these points:

• Draw a straight line through them.
• Ensure the line extends both left and right beyond the points, as lines are infinite in length.

These two intercepts define the line \( y = \frac{x-3}{2} \).
While graphing, visualization of the line as part of an infinite series of points along that line can simplify understanding. This visual guide is vital for anyone learning to graph linear equations, giving a clear visual connection between algebraic solution and graphical representation.