Problem 59 - College Algebra with Modeling and Visualization

Question

Determine the \(x\) - and \(y\) -intercepts on the graph of the equation. Graph the equation. \(x-y=7\)

Step-by-Step Solution

Answer

x-intercept: (7, 0); y-intercept: (0, -7).

Step 1: Identify the Equation

We start with the linear equation given in the problem, which is \(x - y = 7\). Our task is to find both the \(x\)-intercept and the \(y\)-intercept for this equation.

Step 2: Find the x-intercept

To find the \(x\)-intercept, we set \(y = 0\) and solve for \(x\). Substitute \(y = 0\) into the equation: \[x - 0 = 7\]. Simplifying, we get \(x = 7\). Thus, the \(x\)-intercept is the point \((7, 0)\).

Step 3: Find the y-intercept

To find the \(y\)-intercept, we set \(x = 0\) and solve for \(y\). Substitute \(x = 0\) into the equation: \[0 - y = 7\]. Simplifying, we get \(y = -7\). Thus, the \(y\)-intercept is the point \((0, -7)\).

Step 4: Plot the Intercepts on a Graph

Draw a Cartesian plane and plot the \(x\)-intercept \((7, 0)\) and the \(y\)-intercept \((0, -7)\) as points on the graph. These points will help us draw the line.

Step 5: Draw the Line

Using a ruler, draw a straight line through the points \((7, 0)\) and \((0, -7)\). This is the graph of the equation \(x - y = 7\). The line extends infinitely in both directions, but between these intercepts, you have a clear segment of the line.

Key Concepts

Graphing Intercepts

Graphing a linear equation involves plotting its intercepts, which are points where the line crosses the axes. Knowing how to graph using intercepts is fundamental in understanding linear functions. The intercepts are typically easy to find and provide a quick way to sketch a graph without needing additional points.

Here's a simple approach to graphing using intercepts:

• Identify the equation of the line, similar to our example, which is given by the equation: \(x - y = 7\).
• Find the intercepts, the specific points where the line crosses the axes.
• Plot the intercepts on a coordinate plane.
• Draw a line through these intercepts to complete the graph.

By following these steps, the graph of any reasonably straightforward equation can be plotted with ease.

x-intercept

The x-intercept of a graph is the point where the line crosses the x-axis. At this point, the y-coordinate is always zero because it is the horizontal axis.

To find the x-intercept for the equation \(x - y = 7\), we follow these steps:

• Set \(y = 0\) in the equation.
• This simplifies the equation to \(x - 0 = 7\), meaning \(x = 7\).
• Therefore, the x-intercept is the point \((7, 0)\).

It is a crucial part of graphing linear equations as it helps in accurately representing the graph on the coordinate plane.

y-intercept

The y-intercept of a graph occurs where the line crosses the y-axis. Here, the x-coordinate is always zero because it represents the vertical axis.

To find the y-intercept in the equation \(x - y = 7\):

• Set \(x = 0\) in the equation, which becomes \(0 - y = 7\).
• Solving for \(y\), we get \(y = -7\).
• Thus, the y-intercept is located at the point \((0, -7)\).

Identifying the y-intercept is essential for creating a complete and accurate graph of a linear equation.