Problem 67 - Precalculus Essentials

Question

Use intercepts to graph equation. $$6 x-2 y-12=0$$

Step-by-Step Solution

Answer

The x-intercept of the equation is 2 and the y-intercept is -6. By connecting these points on a coordinate plane, a line representing the graph of the equation is obtained.

Step 1: Find the x-intercept (let y=0)

To find the x-intercept, we set y = 0 in the equation and solve for x. This becomes: \[6x - 2(0) - 12 = 0 =>6x = 12 => x = 2\]. Therefore, the x-intercept is 2.

Step 2: Find the y-intercept (let x=0)

To find the y-intercept, we set x = 0 in the equation and solve for y. This means: \[6(0) - 2y - 12 = 0 => -2y = 12 => y = -6\]. Therefore, the y-intercept is -6.

Step 3: Plot the intercepts and draw the line

Now that we have the x-intercept (2, 0) and the y-intercept (0, -6), we plot these points on the coordinate plane. Next, we connect these points using a ruler to make a straight line. This line represents the graph of the equation.

Key Concepts

Understanding the x-intercept

The x-intercept of a linear equation is where the line crosses the x-axis on a coordinate plane. This point has a y-value of zero because it lies on the horizontal axis itself, meaning the equation is solved by finding where the output, or 'y', of the line equals zero.
To find the x-intercept, you substitute 0 for y in the equation and solve for x. For the equation \(6x - 2y - 12 = 0\), setting \(y=0\) gives us the equation \(6x - 12 = 0\).
From this step, solving for x involves a few simple steps:

• Add 12 to both sides to get \(6x = 12\).
• Divide both sides by 6 to isolate x, resulting in \(x = 2\).

Thus, the x-intercept for this equation is the point (2, 0). This means the line will touch the x-axis at the point where x equals 2.

Finding the y-intercept

The y-intercept is the point where the line of the equation crosses the y-axis. At this crossing point, the x-value is always zero because the line reaches the y-axis when there is no movement along the x-axis.
To find the y-intercept, you make x equal to zero in the equation and then solve for y. For the equation \(6x - 2y - 12 = 0\), replacing x with 0 gives us the equation \(-2y - 12 = 0\).
From this point, you solve for y:

• Add 12 to both sides to obtain \(-2y = 12\).
• Divide each side by -2 to solve for y, which gives \(y = -6\).

The y-intercept here is (0, -6). This signifies the line will intersect the y-axis at the point where y equals -6.

Exploring the coordinate plane

A coordinate plane is a two-dimensional plane formed by two perpendicular number lines: the x-axis, which is horizontal, and the y-axis, which is vertical. These axes intersect at a point known as the origin, labeled as (0, 0). Each point on the plane can be identified by an ordered pair of numbers, or coordinates.
The first number, known as the x-coordinate, indicates the position along the x-axis, while the second number, the y-coordinate, specifies the position along the y-axis.
To graph a linear equation, like \(6x - 2y - 12 = 0\), you need at least two points. The x-intercept (2, 0) and y-intercept (0, -6) are perfect choices. Here’s how you plot them:

• Locate the x-intercept: Move 2 units to the right of the origin along the x-axis, without moving up or down.
• Mark the coordinates (2, 0) on the plane.
• Find the y-intercept: Move 6 units below the origin on the y-axis.
• Place the point (0, -6) on the plane.

Once these points are plotted, draw a straight line passing through them; this line represents all the solutions to the linear equation on the coordinate plane.