Problem 33 - Introductory Algebra for College Students

Question

Use intercepts and a checkpoint to graph equation. \(2 x-8 y=12\)

Step-by-Step Solution

Answer

The x-intercept is (6,0) and the y-intercept is (0,-1.5). Draw a line through these points to create the graph of the equation. Choose (6,0) as a checkpoint.

Step 1: Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\). This results in: \(2x - 8*0 = 12\), which simplifies to \(2x = 12\), so \(x = 6\). The x-intercept is thus (6,0).

Step 2: Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\). This results in: \(2*0 - 8y = 12\), simplifying to \(-8y = 12\), which further simplifies to \(y = -12/8 = -1.5\). The y-intercept is thus (0, -1.5).

Step 3: Graph the intercepts

Plot the points (6,0) and (0,-1.5) on the graph. These are the points where the line crosses the x-axis and y-axis, respectively.

Step 4: Draw the graph using intercepts

Draw a straight line that passes through both intercept points. This line represents the graph of the equation.

Step 5: Identify a checkpoint

A checkpoint is any known point on the line. We have already identified two points (the intercepts), so either of these could serve as a checkpoint. Let's choose (6,0) as our checkpoint.

Key Concepts

X-Intercept

The x-intercept of a line on a graph is the point where the line crosses the x-axis. This happens when the value of y is zero. To find the x-intercept algebraically, you set the y-value to 0 in the equation and solve for x.

For the equation given in the exercise, \(2x - 8y = 12\), we find the x-intercept by substituting y with 0, which gives us \(2x = 12\). Dividing both sides by 2, we get \(x = 6\). Therefore, the x-intercept is the point (6, 0), which indicates that the line crosses the x-axis at the point where x is 6.

Y-Intercept

Conversely, the y-intercept is where a line crosses the y-axis, which occurs when the value of x is zero. To locate the y-intercept on the equation \(2x - 8y = 12\), we replace x with 0. This yields \( -8y = 12\), and after dividing each side by -8, it simplifies to \(y = -1.5\). Therefore, the y-intercept is at the point (0, -1.5) on the graph, meaning our line crosses the y-axis at the point where y is -1.5.

The concept of y-intercept is essential as it often represents the starting point of the relationship described by the equation when no input (x) is applied.

Algebraic Graphing

The process of algebraic graphing involves plotting the behavior of an equation onto a coordinate plane. It’s a visual representation of the solutions to the given equation. The steps to graph the equation typically start by identifying the intercepts because they provide easily calculable points through which the line passes.

After plotting the x-intercept and y-intercept, you need at least one more point to ensure the accuracy of the line. This is sometimes called a 'checkpoint'. Given our linear equation, if we choose an arbitrary value for x or y, we can solve for the other variable to locate another point on the line. The intercepts and checkpoint give us a precise idea of the line's slope and direction on the graph.

Linear Equations

A linear equation is an equation involving two variables that produces a straight line when graphed on a coordinate plane. These equations typically take the form \(Ax + By = C\), where A, B, and C are constants. The line's slope and intercepts can be quickly identified with this standard form.

In the context of our exercise, \(2x - 8y = 12\), it's a linear equation since any value of x and y that satisfies the equation falls on a straight line. Graphing linear equations is fundamental in algebra, and understanding their behavior helps in visualizing how variables relate to each other. Learning to graph these equations with precision leads to a deeper comprehension of algebraic concepts and functionalities in real-world applications.