Problem 27 - A Graphical Approach to College Algebra

Question

Graph each line by hand. Give the \(x\)- and y-intercepts. \(x-y=4\)

Step-by-Step Solution

Answer

X-intercept: (4, 0), Y-intercept: (0, -4), Line equation: y = x - 4.

Step 1: Convert to Slope-Intercept Form

Start by rewriting the equation of the line in the slope-intercept form, which is \(y = mx + b\). The given equation is \(x - y = 4\). Solve for \(y\) by subtracting \(x\) from both sides: \(-y = -x + 4\). Multiply the entire equation by \(-1\) to isolate \(y\): \(y = x - 4\). Now you have the slope-intercept form \(y = x - 4\).

Step 2: Determine the Y-Intercept

The y-intercept can be found directly from the slope-intercept form equation \(y = x - 4\). This form is \(y = mx + b\), where \(b\) is the y-intercept. Here, \(b = -4\), so the y-intercept is \( (0, -4) \).

Step 3: Determine the X-Intercept

To find the x-intercept, set \(y = 0\) in the original equation \(x - y = 4\) and solve for \(x\). Substitute \(0\) for \(y\): \(x - 0 = 4\). Simplifying, \(x = 4\). So, the x-intercept is \( (4, 0) \).

Step 4: Plot the Intercepts

On a coordinate plane, plot the intercepts you found. Mark the point \( (0, -4) \) for the y-intercept and \( (4, 0) \) for the x-intercept.

Step 5: Draw the Line

Using a ruler, connect the y-intercept \((0, -4)\) and x-intercept \((4, 0)\) with a straight line. This line represents the graph of the equation \(x - y = 4\). Make sure the line extends across the graph to show its continuous nature.

Key Concepts

Slope-Intercept Form

When it comes to graphing linear equations, the slope-intercept form is one of the most useful formats. This form is written as \(y = mx + b\). Understanding what \(m\) and \(b\) represent is key:

• \(m\) is the slope of the line, which shows how steep the line is. It tells you how much \(y\) changes for a change in \(x\).
• \(b\) is the y-intercept, which shows where the line crosses the y-axis. This is where \(x = 0\).

Let's relate this to the given equation. By converting the equation \(x - y = 4\) into slope-intercept form, we get \(y = x - 4\). Here, the slope \(m\) is 1 (implying the line goes up 1 unit for every 1 unit it goes to the right), and the y-intercept \(b\) is \(-4\). This conversion helps us easily visualize and graph the line.

X-Intercept

The x-intercept of a line is where the line crosses the x-axis. At this point, the value of \(y\) is always 0. To find the x-intercept, you need to substitute \(y = 0\) into the equation and solve for \(x\).
For the equation \(x - y = 4\), setting \(y\) to 0 gives us \(x - 0 = 4\). Simplifying this gives \(x = 4\), which means the x-intercept is at the point \((4, 0)\).
Plotting the x-intercept is crucial when drawing lines by hand. It serves as one of the main reference points to connect with the y-intercept.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis. Here, the value of \(x\) is always 0. In the slope-intercept form \(y = mx + b\), \(b\) represents the y-intercept directly.
For the equation derived in the slope-intercept form \(y = x - 4\), the y-intercept \(b\) is \(-4\). This indicates that the line crosses the y-axis at the point \((0, -4)\).
By plotting the y-intercept on the graph, you have a starting point to draw the line. Connecting this point with the x-intercept enables you to visualize the complete graph of the line, allowing you to see its direction and how it interacts with the axes.