Problem 1 - Thinking Mathematically

Question

In Exercises \(1-8\), use the \(x\) - and \(y\)-intercepts to graph each linear equation. \(x-y=3\)

Step-by-Step Solution

Answer

The x-intercept is 3, the y-intercept is -3, and the line passing through the points (3,0) and (0,-3) is the graph of the given linear equation.

Step 1: Find the x-intercept

To calculate the x-intercept, set \(y = 0\) in the original equation and solve for \(x\). Substituting \(y = 0\) in the equation \(x - y = 3\), we get \(x = 3\). So, the x-intercept is 3.

Step 2: Find the y-intercept

To calculate the y-intercept, set \(x = 0\) in the original equation and solve for \(y\). Substituting \(x = 0\) in the equation \(x - y = 3\), we get \(-y = 3\), so \(y = -3\). Thus, the y-intercept is -3.

Step 3: Plot the graph

Plot the x- and y-intercepts on a graph. The x-intercept (3,0) and the y-intercept (0,-3) are two points on the line. Draw a line passing through these points. This is the graph of the given linear equation.

Key Concepts

Understanding X-Intercepts

In the world of linear equations, the **x-intercept** is where the line crosses the x-axis. This means at the x-intercept, the value of y is zero. Understanding x-intercepts is crucial because they help us to see how the line behaves on a graph. They are easy to find. You just set the y-value to zero in the equation and solve for x.

For example, let’s look at the equation given:

• Substitute y = 0 into the equation, which is: \( x-y=3 \).
• This becomes \( x-0=3 \) simplifying to \( x=3 \).

So, the x-intercept of the equation is 3. This tells us that the line passes through the point (3, 0) on the graph. Knowing where the line crosses the x-axis can assist in understanding the bigger picture of the graph.

Exploring Y-Intercepts

The **y-intercept** represents the point where the line crosses the y-axis. At this intercept, the x-value is always zero. Like the x-intercept, it's essential in figuring out both the slope and the position of the line on the graph. To find the y-intercept, you simply substitute 0 for x and solve for y.

Following the example of the given equation:

• Set x = 0 in the equation: \( x - y = 3 \).
• This changes the equation to \( 0 - y = 3 \), which simplifies to \( y = -3 \).

Thus, the y-intercept is -3, indicating the line crosses the y-axis at the point (0, -3). Finding the y-intercept is a key step in sketching the graph of the linear equation on paper or mentally.

Graphing Linear Equations

Graphing linear equations is a visual way to see the relationship between variables. It helps in understanding how the change in one quantity affects another. Once you have the intercepts, plotting the graph becomes straightforward. Just plot the two points—the x-intercept and the y-intercept—and draw a line through them.

Here’s how you can graph the equation \( x - y = 3 \):

• Plot the x-intercept at (3, 0) on the Cartesian plane.
• Then, plot the y-intercept at (0, -3).
• Once both points are plotted, draw a straight line through these points.

This line represents the set of all solutions to the linear equation \( x - y = 3 \). Graphing helps us directly visualize the solutions and better grasp the nature of linear equations. It’s a fundamental skill that gives insights not only in mathematics but also in fields dealing with linear relationships.