Problem 37 - Introductory Algebra for College Students

Question

Graph each linear equation using the slope and y-intercept. $$y=-\frac{5}{3} x$$

Step-by-Step Solution

Answer

The graph represents the equation \(y = -\frac{5}{3}x\), with a slope of -5/3 and y-intercept of 0. It crosses the origin and falls from left to right at a constant rate of 5 vertical units for every 3 horizontal units.

Step 1: Identify the slope and y-intercept

The equation of the line is in the form \(y = mx + c\), where m is the slope and c is the y-intercept. In the given equation \(y = -\frac{5}{3}x\), the slope m is -\frac{5}{3} and the y-intercept c is 0.

Step 2: Plot the y-intercept

The first point to plot is the y-intercept. Since the y-intercept is 0, plot a dot at point (0,0) on the graph.

Step 3: Plot another point using the slope

From the y-intercept, move along the line of the graph according to the slope. The slope is -\frac{5}{3}, meaning for every 3 units moved horizontally (run), move 5 units downwards (falling slope is indicated by the negative sign) vertically. This gives a second point to be marked on the graph.

Step 4: Draw the line

Finally, draw a straight line through the two plotted points. This line represents the equation \(y = -\frac{5}{3}x\).

Key Concepts

Slope and Y-Intercept

When graphing linear equations, understanding the slope and y-intercept is crucial. The slope indicates the steepness of the line and its direction. It is calculated as the ratio of the rise (vertical change) to the run (horizontal change). A positive slope means the line inclines upward as it moves from left to right, whereas a negative slope, as in the equation \(y = -\frac{5}{3}x\), shows that the line declines.

The y-intercept is where the line crosses the y-axis. This is the point \( (0, c) \), where \( c \) is the y-intercept value in the slope-intercept form of a linear equation \(y = mx + c\). In our exercise, the y-intercept is \(0\), indicating that the line goes through the origin. Learning to correctly identify these two features is essential to plot the graph accurately.

Plotting Points

To graph a linear equation, plotting points effectively is key. Start with the y-intercept, which is the simplest point to locate. In our example, the y-intercept is at the origin, which is \( (0, 0) \). After plotting this point on the graph, you need to find at least one more point to draw the line. Using the slope \( -\frac{5}{3} \) helps determine the second point. Since the slope is negative, you move to the right and then down, reflecting the negative direction. You move 3 units right (the run) and 5 units down (the rise). The coordinates of the new point would be \( (3, -5) \). By plotting and connecting these points with a straight line, you confirm the graphical representation of your equation.

Slope-Intercept Form

The slope-intercept form, \(y = mx + c\), is the most commonly used format to express a linear equation because it clearly displays the slope \(m\) and y-intercept \(c\). This form makes graphing simpler, as you can immediately identify these key components. For the given equation \(y = -\frac{5}{3}x\), the slope-intercept form indicates a slope \(m\) of \( -\frac{5}{3} \) and a y-intercept \(c\) of \(0\). It's beneficial to rewrite any linear equation into this form before starting the graphing process to streamline your work.

Linear Equation Representation

A linear equation can be represented graphically by a straight line on the Cartesian plane. Every point on that line satisfies the equation. To represent our example \(y = -\frac{5}{3}x\) as a graph, you would start by plotting the y-intercept, followed by additional points based on the slope, and then connect these to form a line. The slope dictates how sharply the line angles and in which direction, while the y-intercept specifies where it crosses the y-axis. A proper understanding of the linear equation representation through graphical methods offers a visual way to comprehend the behaviors and characteristics of the line, such as its steepness, direction, and the points through which it passes.