Question

In order to complete Parts 1 and 2, you will be graphing two linear equations. Create a separate coordinate plane for each Part 1 and Part 2 or print two separate pieces of graph paper. Be sure to include your answers for all of Parts 1-3.

Determine the X- and y-Intercepts of the line created by the linear equation 3x - y - 1 = 0. Use the Intercepts to graph the line. On your graph, label the points (coordinates) that represent the X- and y-Intercepts.

Determine the slope and y-Intercept of the linear equation 3x - y - 1 = 0. Use the slope and y-Intercept to graph the line. On your graph, label the point (coordinate) that represents the y-Intercept and one additional point (coordinate) found by applying the slope of the line.

In the form of a paragraph, compare and contrast the methods used for graphing the linear equation, 3x - y - 1 = 0, in Parts 1 and 2."

Answer 1

The problem consists of finding the line's intercepts and using them to graph it, as well as determining the slope and y-intercept to graph it by another method. Both methods graph the same line, but with different initial steps.

Step-by-step explanation:

The problem involves determining the x-intercept and y-intercept of the line represented by the equation 3x - y - 1 = 0. To find the x-intercept, we set y to 0 and solve for x, which yields x = 1/3. To find the y-intercept, we set x to 0 and solve for y, which yields y = -1. Hence, the intercepts are (1/3, 0) and (0, -1).

For the second part, we need to determine the slope and y-intercept of the same equation for graphing. We first write the equation in slope-intercept form y = mx + b, which translates to y = 3x + 1. Here, the slope (m) is 3, and the y-intercept (b) is 1. Starting from the y-intercept (0, 1), we apply the slope to find another point by rising 3 units in the vertical direction for every 1 unit we move to the right on the horizontal axis. Labeling the y-intercept and the additional point found using the slope provides enough information to graph the line.

Comparing the methods, using intercepts allows us to find two precise points where the line crosses the axes, effectively defining the line. Applying the slope and y-intercept method involves starting with the point where the line crosses the y-axis and using the slope to find another point. Both methods will render the same line on a graph, but the intercept approach directly provides the x and y bounds where the line cuts the axes, while the slope-intercept method emphasizes the rate of change and starting value of the function.