Problem 34 - Thomas Calculus

Question

In Exercises \(31-34,\) find the line's \(x\) - and \(y\) -intercepts and use this information to graph the line. $$ 1.5 x-y=-3 $$

Step-by-Step Solution

Answer

The x-intercept is -2, and the y-intercept is 3.

Step 1: Rearrange Equation

To find the intercepts, first, let's rearrange the equation to solve for each variable by itself when the other is zero. The given equation is:\[1.5x - y = -3\]

Step 2: Find the X-Intercept

To find the x-intercept, set \(y = 0\) and solve for \(x\):\[1.5x - 0 = -3\]\[1.5x = -3\]\[x = \frac{-3}{1.5} = -2\]Thus, the x-intercept is at \((-2, 0)\).

Step 3: Find the Y-Intercept

To find the y-intercept, set \(x = 0\) and solve for \(y\):\[1.5(0) - y = -3\]\[-y = -3\]\[y = 3\]Thus, the y-intercept is at \((0, 3)\).

Step 4: Graph the Line

Now use the intercepts to graph the line. Plot the points \((-2, 0)\) and \((0, 3)\) on the coordinate plane and draw a straight line through these points. This line represents the equation \(1.5x - y = -3\).

Key Concepts

Intercepts

Finding intercepts is crucial when dealing with linear equations. They tell us where the line crosses the x-axis and y-axis on the coordinate plane. Let’s look at what they mean and how to find them:

• The x-intercept is the point where the line crosses the x-axis. At this point, the value of y is zero.
• The y-intercept is the point where the line crosses the y-axis, and here, the value of x is zero.

To find these intercepts from the equation of a line given by \(1.5x - y = -3\), start by setting the opposite variable to zero and solve for the remaining one.

- For the x-intercept, set \(y = 0\) and solve: \[1.5x = -3\] Solving gives \(x = -2\). So, the point is \((-2, 0)\).

- For the y-intercept, set \(x = 0\) and solve: \[-y = -3\] Solving gives \(y = 3\). So, the point is \((0, 3)\).
Knowing these key intercepts is a big step in visualizing and graphing a line correctly.

Graphing Lines

Graphing lines with intercepts is straightforward and helpful for understanding linear equations.
When you have both the x- and y-intercepts, you can draw a line effortlessly on the coordinate plane.

• Start by plotting the x-intercept on the graph. In our case, mark the point \((-2, 0)\) on the x-axis.
• Next, plot the y-intercept \((0, 3)\) on the y-axis.

Use these two points to draw a straight line across the graph. This line is the graphical representation of the equation \(1.5x - y = -3\).

Remember, for accuracy:

• Use a ruler or a straight edge to connect the points.
• Make sure to extend the line beyond the two points in both directions, as lines continue indefinitely.

This step-by-step approach allows you to visualize the behavior of the line and see its relationship with the axes.

Coordinate Plane

Understanding a coordinate plane is essential for graphing not just lines, but various functions.
A coordinate plane consists of a horizontal axis known as the x-axis and a vertical axis known as the y-axis.
It’s where you plot points using ordered pairs \((x, y)\). Each point tells you how far along each axis the point is located.

• The origin is the center of the coordinate plane at \((0, 0)\), where the two axes intersect.
• The plane is divided into four sections called quadrants, labeled counterclockwise: I, II, III, and IV.
• In Quadrant I, both x and y are positive, while in Quadrant III, both are negative.

When graphing the line from our exercise using the equation \(1.5x - y = -3\), understanding the coordinates and how they are oriented enables you to accurately place intercepts and draw the line's path.

With practice, visualizing lines on this grid becomes intuitive, giving you a deeper insight into linear equations and their applications.