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Chapter 1: Problem 58

Use a graphing utility to graph the equation. Use a standard setting. Approximate any intercepts. $$y=\frac{2}{3} x-1$$

Short Answer

Expert verified
The graph of the equation \(y = \frac{2}{3}x - 1\) is a straight line with a slope of \(\frac{2}{3}\). The y-intercept is -1 and the x-intercept is approximately 1.5.

Step by step solution

01

Determine the Slope and Y-Intercept

The equation given is \(y = \frac{2}{3}x - 1\). Here, \(\frac{2}{3}\) is the slope, and -1 is the y-intercept (the value of y when x = 0).
02

Plot Y-Intercept

The Y-intercept is -1, i.e., the point where the line crosses the Y-axis is (0, -1). Start by plotting this point on the graph.
03

Use Slope to Locate another Point

The slope \(\frac{2}{3}\) can be interpreted as a rise of 2 (positive direction) for a run of 3 (positive direction). From the y-intercept (0, -1), move up 2 units and right 3 units to get to the next point (3, 1) on the line. Plot this point on the graph.
04

Draw the Line

Now, create a straight line that passes through these two points.
05

Identify and approximate intercepts

Intercepts are points where the line crosses the x or y axes. You have already identified the y-intercept as -1. The x-intercept is where the line crosses the x-axis (when y = 0). To find this, set \(y = 0\) in the equation and solve for \(x\): \(0 = \frac{2}{3}x - 1\), which simplifies to \(x =\frac{3}{2}\). So the x-intercept is approximately \(1.5\). So the intercepts are approximately (1.5, 0) and (0, -1).

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