Problem 66 - Introductory Algebra for College Students

Question

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation. $$y=-3 x-2$$

Step-by-Step Solution

Answer

The graph of the equation \(y = -3x - 2\) passes through the points (-1,1), (0,-2), (1, -5), (2,-8), and (3,-11)

Step 1: Understand the given equation

The equation given here is \(y = -3x - 2\) which is in the form \(y = mx + c\), where \(m\) represents the slope and \(c\) is the y-intercept.

Step 2: Make a Table of Values

A table of values is a tool used to generate points on a graph. For the equation \(y = -3x - 2\), choose five distinct values for \(x\) and calculate the corresponding \(y\) values.\n\n For \(x = -1, 0, 1, 2, 3\), the corresponding \(y\) values are \(1, -2, -5, -8, -11\) respectively.

Step 3: Plot the Points and Draw the Line

Plot the points from the table of values on a graph and draw a straight line passing through them. This line represents the solutions to the given linear equation.

Key Concepts

Table of Values

A table of values is a simple and effective tool for graphing linear equations. It allows you to see how changes in one variable affect the other. To create a table of values:

• Choose several values for the independent variable, usually denoted as \(x\).
• Substitute each \(x\) value into the equation to find the corresponding \(y\) value.

For example, given the equation \(y = -3x - 2\):

• If \(x = -1\), then \(y = -3(-1) - 2 = 1\).
• If \(x = 0\), then \(y = -3(0) - 2 = -2\).
• If \(x = 1\), then \(y = -3(1) - 2 = -5\).
• If \(x = 2\), then \(y = -3(2) - 2 = -8\).
• If \(x = 3\), then \(y = -3(3) - 2 = -11\).

With these \(x\) and \(y\) pairs, you can now proceed to plot the points on a graph.

Slope

The slope of a line is a measure of how steep the line is. It tells you how much \(y\) changes for every one unit change in \(x\).
The equation of the line is \(y = -3x - 2\). The slope here is \(-3\), which means:

• For every one unit increase in \(x\), \(y\) decreases by 3 units.

The slope indicates a negative direction, given by the minus sign. A negative slope means the line falls as you move from left to right across the graph.
Understanding slope helps you immediately identify how the graph behaves without plotting many points.

Y-Intercept

In the equation of a line, the y-intercept is the point where the line crosses the y-axis. It is represented by the constant term in the equation.
For \(y = -3x - 2\), the y-intercept is \(-2\). This means:

• The line crosses the y-axis where \(y = -2\) and \(x = 0\).

The y-intercept provides a useful starting point when you're plotting the graph. It helps confirm the line's position even before plotting other points.

Plotting Points

Plotting points is a crucial step in graphing a linear equation. Once you've calculated your table of values, you're ready to place these points on a graph.
To plot the points:

• Draw a set of axes on graph paper or use a graphing tool.
• Locate each \(x\) and \(y\) coordinate pair on the graph.
• For instance, plot \((-1, 1)\), \((0, -2)\), \((1, -5)\), \((2, -8)\), and \((3, -11)\).
• After plotting all the points, draw a straight line through them. This line is the graphical representation of your equation.

The line should pass through all plotted points if your calculations are correct. By practicing plotting points, you gain insights into the relationship between the algebraic equation and its visual representation.