Problem 29 - A Graphical Approach to College Algebra

Question

Graph each line by hand. Give the \(x\)- and y-intercepts. \(3 x-y=6\)

Step-by-Step Solution

Answer

The y-intercept is (0, -6); the x-intercept is (2, 0).

Step 1: Identify the Standard Form

The given equation is in the form of a linear equation: \(3x - y = 6\). Guide through converting this into a more graph-friendly form if needed, but recognize its standard form first.

Step 2: Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation \(3x - y = 6\): \[3(0) - y = 6 \y = -6\]Hence, the y-intercept is (0, -6).

Step 3: Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation \(3x - y = 6\):\[3x - 0 = 6 \x = 2\]Thus, the x-intercept is (2, 0).

Step 4: Plot the Intercepts

On a graph, plot the intercepts found: (0, -6) on the y-axis and (2, 0) on the x-axis.

Step 5: Draw the Line

Using a ruler, draw a straight line through the plotted intercepts (0, -6) and (2, 0). This line represents the equation \(3x - y = 6\).

Key Concepts

Understanding the x-intercept

The x-intercept of a line is the point where the line crosses the x-axis. This occurs when the value of y is zero. To find the x-intercept, substitute 0 for y in the equation, and solve for x. In our example, we start with the equation: \[3x - y = 6\] By setting y to 0, we simplify the equation to: \[3x - 0 = 6\] Solving for x, we divide both sides by 3 to get: \[x = 2\] Therefore, the x-intercept is at the point (2, 0). This means the line will touch the x-axis at this point. Understanding the location of the x-intercept is crucial, as it is one of the two primary points used to graph a straight line using intercept methods.

Exploring the y-intercept

The y-intercept is the point where the line crosses the y-axis. This happens when x is equal to zero. To determine the y-intercept, make x equal to 0 in the line's equation and solve for y. For the equation: \[3x - y = 6\] we set x to 0, leading to the simplification: \[3(0) - y = 6\] The equation simplifies to: \[-y = 6\] Solving for y involves multiplying both sides by -1, resulting in: \[y = -6\] The y-intercept is therefore at the point (0, -6). On the graph, this means the line will intersect the y-axis at this point. Recognizing the y-intercept provides one of the coordinates needed to draw the line on a graph.

Steps in graphing lines using intercepts

Graphing lines can be simple when using the x- and y-intercepts. Here's how you can quickly graph a line using these two points:

• Identify the x-intercept by setting y to 0 in the equation and solving for x.
• Identify the y-intercept by setting x to 0 in the equation and solving for y.
• Plot both of these points on the graph.

Once both intercepts are marked:- Connect these points using a ruler to ensure your line is straight.- Extend the line beyond the intercepts if needed to show a complete linear graph.This method works well for linear equations and provides a quick way to visualize the relationship represented by the equation. In our example, after plotting (2, 0) and (0, -6), drawing a line through these points represents the graph of the equation \(3x - y = 6\). This clear visual can often make understanding the equation more intuitive.