Problem 68 - 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. $$y=-\frac{1}{2} x$$

Step-by-Step Solution

Answer

For the given equation \( y = -\frac{1}{2}x \), the following solution pairs (x, y) are obtained: (0, 0), (2, -1), (4, -2), (6, -3), and (8, -4). These values are obtained by substituting x-values into the equation and calculating for y. The line drawn on the graph passes through these points.

Step 1: Understanding the Linear Equation

The given linear equation is \( y = -\frac{1}{2}x \). This is in the slope-intercept form where the slope \( m = -\frac{1}{2} \) and the y-intercept \( c = 0 \).

Step 2: Substituting x-values and calculating for y-values

Substitute suitable x-values into the equation and calculate the corresponding y-values. A table is drawn for easier visualization.

Step 3: Plotting the Graph

Using the values from the table obtained in step 2, plot points for each solution pair (x, y) on the graph. Join these points using a straight line.

Key Concepts

Understanding Slope-Intercept Form

When graphing linear equations, one of the most common formats used is the slope-intercept form. In its simplest terms, the slope-intercept form of a line is given by the equation \( y = mx + b \) where \( m \) is the slope of the line and \( b \) represents the y-intercept, which is the point where the line crosses the y-axis.

Let's decipher the line \( y = -\frac{1}{2}x \) from our exercise. Here, the slope \( m \) is \( -\frac{1}{2} \) which tells us that for every unit we move to the right along the x-axis, we move down half a unit on the y-axis. The y-intercept \( b \) is \( 0 \) in this case, indicating that our line crosses the y-axis right at the origin (0,0). With these components, we have full knowledge of how our line behaves and can proceed to graph it effectively.

Creating a Table of Values

A table of values is used to represent ordered pairs \( (x, y) \) of a linear equation, providing a clear overview of the points that lie on the graph of the line. To create a table of values, select a range of x-values (both positive and negative) and then use the slope-intercept form to solve for their corresponding y-values.

For our exercise, choosing x-values like -2, -1, 0, 1, and 2, we substitute each into the equation to find y. As an example, when \( x = -2 \) then \( y = -\frac{1}{2}(-2) = 1 \). Repeat this process with the other x-values to fill out your table. This method presents a simple and systematic approach to finding points that you can then plot on a graph to visualize your linear equation.

Plotting Points and Graphing the Line

With a complete table of values, the next step is to plot the points on a coordinate plane. By plotting each ordered pair, you give a visual outline of where the graph of your equation will pass. Start with the y-intercept, then plot the remaining points using the x and y values from your table.

In the context of our exercise, after plotting all the points from the table, connect them with a straight line that extends in both directions past the plotted points. This line represents the complete set of solutions for the equation \( y = -\frac{1}{2}x \). Remember, the accuracy of your line depends on your points, so double-check your table of values and the precision of your plotting for the best result.