Problem 39 - College Algebra with Corequisite Support

Question

For each of the following exercises, find and plot the \(x\) - and \(y\) -intercepts, and graph the straight line based on those two points. $$ x-2 y=8 $$

Step-by-Step Solution

Answer

The x-intercept is (8, 0), the y-intercept is (0, -4), and the line is drawn through these points.

Step 1: Identify the Equation Type

The given equation \( x - 2y = 8 \) is a linear equation. It represents a straight line in a two-dimensional plane.

Step 2: Find the x-intercept

To find the \( x \)-intercept, set \( y = 0 \) in the equation and solve for \( x \):\[ x - 2(0) = 8 \ x = 8\]Hence, the \( x \)-intercept is at \( (8, 0) \).

Step 3: Find the y-intercept

To find the \( y \)-intercept, set \( x = 0 \) in the equation and solve for \( y \):\[ 0 - 2y = 8 \ -2y = 8 \ y = -4\]Thus, the \( y \)-intercept is at \( (0, -4) \).

Step 4: Plot the Intercepts

Plot the points \( (8, 0) \) and \( (0, -4) \) on a Cartesian plane. These points represent the \( x \)-intercept and \( y \)-intercept of the line.

Step 5: Draw the Line

Draw a straight line through the intercept points \( (8, 0) \) and \( (0, -4) \) to complete the graph of the equation \( x - 2y = 8 \).

Key Concepts

x-intercept

In mathematics, the **x-intercept** of a line is the point where the line crosses the x-axis of a Cartesian coordinate plane. This happens when the y-coordinate is zero because all points along the x-axis have a y value of zero.
Finding the x-intercept is straightforward. Set the y-value in your linear equation to zero and solve for x. For the equation \( x - 2y = 8 \), when substituting \( y = 0 \), it simplifies to \( x = 8 \). Hence, the x-intercept here is the point \( (8, 0) \).
**Key points to remember about the x-intercept**:

• It's found by setting \( y = 0 \) in the equation.
• It tells you where the line crosses the x-axis.
• It's useful for graphing lines quickly by marking off one key point on the graph.

y-intercept

The **y-intercept** is the point where a line crosses the y-axis. This occurs when the x-coordinate is zero because all points on the y-axis have an x value of zero.
To find the y-intercept, set the x-value in the equation to zero and solve for y. For the equation \( x - 2y = 8 \), substituting \( x = 0 \) gives \( -2y = 8 \), and solving for y, we get \( y = -4 \). Thus, the y-intercept is the point \( (0, -4) \).
**Key traits of the y-intercept include**:

• It is determined by setting \( x = 0 \).
• Indicates where the line will cross the y-axis.
• Helps establish a second point when graphing a line.

graphing

Once you have both the x-intercept and y-intercept, you can easily graph the line. Graphing linear equations visually represents the relationship expressed in the equation on a Cartesian plane.
**Steps for graphing a linear equation**:

• First, plot the x-intercept point on the graph. From the step-by-step solution, we have the x-intercept at \( (8, 0) \).
• Next, plot the y-intercept point, which from the given equation is \( (0, -4) \).
• Draw a straight line through these two points. This line is the graph of the equation \( x - 2y = 8 \).

Graphing is crucial not only for visualizing solutions but also for verifying algebraic solutions by checking if the plotted line accurately represents the equation given. It offers a clear picture of the slope, position, and intersection points of the line.