Problem 71 - College Algebra

Question

Use intercepts to graph each equation. \(8 x-2 y+12-0\)

Step-by-Step Solution

Answer

The x-intercept of the equation is -1.5 and the y-intercept is -6.

Step 1: Simplify the Equation

First, simplify the given equation. The equation is \(8x - 2y + 12 = 0\) after organizing it.

Step 2: Find the x-intercept

The x-intercept is found by setting y to 0 in the equation. So, plug y=0 into the equation to get: \(8x + 12 = 0\). Solving for x gives: \(x = -12 / 8 = -1.5\). Therefore, the x-intercept is -1.5.

Step 3: Find the y-intercept

The y-intercept is found by setting x to 0 in the equation. So, plug x=0 into the equation to get: \(-2y + 12 = 0\). Solving for y gives: \(y = 12 / -2 = -6\). Therefore, the y-intercept is -6.

Step 4: Graph the Line

Plot the x-intercept (-1.5,0) and the y-intercept (0,-6) on the coordinate plane. Then, draw a line through these two points to graph the equation.

Key Concepts

X-Intercept

Understanding the x-intercept is crucial when graphing linear equations. The x-intercept represents the point on the graph where the line crosses the x-axis. To find this point, you set the y-value to zero and solve for x. For instance, in the equation
\(8x - 2y + 12 = 0\),
when you substitute y with 0, you simplify the equation to find the value of x that makes the equation true. This gives you the x-intercept, which, as calculated, is
\(-1.5\).
This point is written as an ordered pair
\((-1.5, 0)\),
indicating that the line touches the x-axis 1.5 units left of the origin.

Y-Intercept

The y-intercept is another fundamental concept and indicates where the line meets the y-axis. To find it, you set the x-value to zero and solve for y. This is because the y-intercept is the point where your line will cross the y-axis, which naturally has no x-value. Using our example equation,
\(8x - 2y + 12 = 0\),
setting x to zero and solving for y gives you a y-intercept of
\(-6\). This is represented as
\((0, -6)\),
placing the line six units below the origin on the y-axis.

Coordinate Plane

The coordinate plane is a two-dimensional surface formed by two number lines that intersect at a right angle. The horizontal axis is known as the x-axis, and the vertical axis is called the y-axis. The point of intersection of these axes is known as the origin. Each point on the plane is defined by an ordered pair of numbers,
\((x, y)\),
representing the coordinates. As illustrated in graphing the equation from our exercise, the points
\((-1.5, 0)\) and
\((0, -6)\),
are plotted on this plane to help visualize the line defined by the original equation. By understanding the coordinate plane, students can more easily graph equations and interpret graphed functions.

Algebraic Manipulation

To graph a linear equation efficiently, algebraic manipulation skills are essential. This process involves rearranging equations to solve for a variable. Techniques include combining like terms, adding or subtracting terms on both sides, multiplying or dividing both sides by a number, and isolating the desired variable. In our example,
\(8x - 2y + 12 = 0\),
to find the x- and y-intercepts, we applied algebraic manipulation by setting one variable to zero and solving for the other. Mastery of these methods streamlines the process of finding intercepts and simplifies the steps needed to graph equations correctly.