Problem 67 - College Algebra

Question

Use intercepts to graph each equation. \(6 x-2 y-12=0\)

Step-by-Step Solution

Answer

The x-intercept is 2 and the y-intercept is -6. After plotting these two points on the graph a straight line drawn through them gives the graph of the equation.

Step 1: Calculate the x-intercept

First, we set y = 0 in the equation \(6x - 2(0) - 12 = 0\) to find the x-intercept. Simplifying gives us the equation as \(6x = 12\). Dividing both sides by 6 yields \(x = 2\). So, the x-intercept is 2.

Step 2: Calculate the y-intercept

Next, we set x = 0 in the equation \(6(0) - 2y - 12 = 0\) to find the y-intercept. Simplifying gives us the equation as \(-2y = 12\). Dividing both sides by -2 yields \(y = -6\). So, the y-intercept is -6.

Step 3: Graph the line

Plot the x-intercept at point (2,0) and the y-intercept at point (0,-6) on the Cartesian Plane. Next draw a straight line through these two points. This line represents the graph of the equation.

Key Concepts

Graphing Linear Equations

To graph a linear equation like \(6x - 2y - 12 = 0\), it can be helpful to use the intercepts method. Linear equations often appear in the general form \(Ax + By + C = 0\). This form is convenient for identifying intercepts, which helps in graphing. Here's a simple process to follow:

• Identify the x-intercept: Set \(y\) to 0 and solve for \(x\).
• Identify the y-intercept: Set \(x\) to 0 and solve for \(y\).
• Plot these intercepts on a graph as the starting points of the line.
• Draw a straight line through these points.

Graphing linear equations using intercepts provides a clear visualization. The straight line you draw represents all solutions to the equation. Each point on the line is a pair \((x, y)\) that satisfies the equation. This method is simple and efficient for quickly sketching graph lines.

x-intercept

The x-intercept is where the graph crosses the x-axis. Here, the value of \(y\) is always 0. To find the x-intercept in a linear equation, follow these steps:

• Substitute \(y = 0\) into the equation.
• Solve the equation for \(x\).

For example, in the equation \(6x - 2y - 12 = 0\), plugging \(y = 0\) simplifies the equation to \(6x = 12\), thus \(x = 2\). This means the x-intercept is at the point \((2, 0)\).

Understanding x-intercepts is crucial because intercepts provide fixed points through which the line passes, helping to accurately draw the line. They are particularly useful in identifying key characteristics of the line on the Cartesian plane.

y-intercept

The y-intercept is the point where the line crosses the y-axis, meaning \(x\) is 0 at this point. To determine the y-intercept from a given equation, you can follow these steps:

• Set \(x = 0\) in the equation.
• Solve the equation for \(y\).

For instance, substituting \(x = 0\) in the equation \(6x - 2y - 12 = 0\) reduces it to \(-2y = 12\). Solving for \(y\) gives \(y = -6\). Thus, the y-intercept is \((0, -6)\).

The y-intercept is an essential component when graphing linear equations because it serves as another anchor point for the line. Alongside the x-intercept, it allows you to draw a precise line representation of the equation on a graph.