Problem 67 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The graph of the equation passes through the x-intercept (2,0) and the y-intercept (0,-6).

Step 1: Finding the x-intercept

Set y = 0 in the given equation and solve for x. This gives: \[6x-2(0)-12=0\] => \[6x-12=0\] => from this \[x = 2\] is obtained, which means the x-intercept is at point (2,0).

Step 2: Finding the y-intercept

Set x = 0 in the given equation and solve for y. This gives: \[6(0)-2y-12=0\] => \[-2y-12=0\] => from this, \[y = -6\] is obtained, which means the y-intercept is at point (0,-6).

Step 3: Plotting the graph

Plot both intercept points on a graph. Draw a line through both points. That's the graph of the equation.

Key Concepts

X-Intercept

The x-intercept of a graph is the point where the line crosses the x-axis. In other words, it's where the value of y is zero. To find the x-intercept from an equation, we set the y variable to zero and solve for x. For example, with the equation 6x - 2y - 12 = 0, we find the x-intercept by calculating 6x - 2(0) - 12 = 0, which simplifies to 6x - 12 = 0. Solving this gives us x = 2, meaning the x-intercept is the point (2, 0).

Visualizing the x-intercept on a graph helps in understanding how a function behaves in relation to the x-axis. Remember, at the x-intercept, the y value will always be zero, regardless of the equation.

Y-Intercept

Conversely, the y-intercept is where a graph crosses the y-axis, or the spot where x equals zero. To find it in an equation, set the x equal to zero and solve for y. Considering the same equation, 6x - 2y - 12 = 0, we calculate it with x as zero: 6(0) - 2y - 12 = 0 leading to -2y - 12 = 0. Consequently, simplifying this yields y = -6, indicating that the y-intercept is at the point (0, -6).

The y-intercept provides a critical starting point when graphing a line as it's the first point you plot before extending the line through other points. It also indicates the value where the graph will intersect the y-axis when all other variables are set to zero.

Plotting Points on a Graph

Once the intercepts are known, plotting the points on a graph is straightforward. Start with a set of axes labeled 'x' and 'y'. Locate the x-intercept along the x-axis; in our example, this is the point (2,0). Next, find the y-intercept on the y-axis, which is (0,-6) for the equation given. Mark these points on the respective axes.

After marking the intercepts, draw a straight line connecting them — this line represents the graph of the equation. It's essential to ensure the line continues beyond the plotted points and extends to the edges of the graph on both ends. This visual representation provides insight into the behavior and solutions of linear equations and is an essential skill in understanding algebra and calculus.