Problem 67 - Introductory and Intermediate Algebra for College Students 4th

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

The five solutions are: (-2, -1), (-1, -0.5), (0, 0), (1, 0.5), and (2, 1). The graph is a straight line passing through these points.

Step 1: Understand the Equation

The given equation is y = 0.5x. It is already in the slope-intercept form of a linear equation, y = mx + b, where m is the slope and b is the y-intercept. In this case, m = 0.5 and there is no explicit y-intercept, so b = 0.

Step 2: Create a Table of Values

For the table of values, select five arbitrary x-values. For simplicity, let's use -2, -1, 0, 1, and 2. These will be the first coordinates of the solutions.

Step 3: Compute Corresponding y-values

Substitute each x-value into the equation to calculate its corresponding y-value. These will be the second coordinates of the solutions. The calculations are as follows: \nFor x = -2, y = 0.5*(-2) = -1. So, the solution is (-2, -1). \nFor x = -1, y = 0.5*(-1) = -0.5. So, the solution is (-1, -0.5). \nFor x = 0, y = 0.5*0 = 0. So, the solution is (0, 0). \nFor x = 1, y = 0.5*1 = 0.5. So, the solution is (1, 0.5). \nFor x = 2, y = 0.5*2 = 1. So, the solution is (2, 1).

Step 4: Plot the Points and Draw the Graph

Now, plot the solutions on a graph of the xy-plane. This means plotting the following points: (-2, -1), (-1, -0.5), (0, 0), (1, 0.5), and (2, 1). All these points should lie on a straight line (since it's a linear equation). Then, draw a straight line through these points to complete the graph of the equation.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is given by the equation \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) indicates the y-intercept, which is the point where the line crosses the y-axis. In the given exercise, the equation \( y = \frac{1}{2}x \) is already in slope-intercept form, with a slope (m) of \( \frac{1}{2} \) and a y-intercept (b) of 0. This implies that the line crosses the origin \( (0,0) \) and rises half a unit on the y-axis for every one unit it moves to the right on the x-axis.

Understanding the slope-intercept form is crucial because it readily shows the steepness of the line and where the line intercepts the y-axis, allowing for a quick sketch of the graph. Additionally, this form is particularly useful when solving for the y-value given an x-value, making it a starting point for creating a table of values.

Table of Values

A table of values is a tool used to organize the x-values and their corresponding y-values computed from the equation of a line. It aids in identifying points that satisfy the linear equation, which can then be plotted on a graph. For the equation \( y = \frac{1}{2}x \), a table of values was constructed using the x-values -2, -1, 0, 1, and 2.

To fill out the table, we substitute each x-value into the equation to find the resulting y-value. This gives us ordered pairs which are the points that lie on the line when graphed. This table is fundamental in graphing as it provides a clear visual representation of the input-output relationship and ensures we have accurate points to plot.

Linear Equation Solutions

The solutions to a linear equation are the set of all possible ordered pairs \( (x,y) \) that make the equation true. These pairs represent points on the graph of the equation that form a straight line. In our exercise, we have found five such solutions by using different x-values and solving for the corresponding y-values. These solutions represent specific points on the line: (-2, -1), (-1, -0.5), (0, 0), (1, 0.5), and (2, 1).

It's important to recognize that there is an infinite number of solutions to a linear equation because you can choose any real number for x and find its corresponding y. However, for the practical purposes of plotting, a handful of solutions that are conveniently spaced apart will usually be sufficient to draw an accurate graph.

Plotting Points

Plotting points on a coordinate grid is the process of marking the exact location of the points which are the solutions to the linear equation. In our exercise, after calculating the y-values for our selected x-values, we get the points that we then identify on the graph. The point (-2, -1), for instance, means that we start at the origin and move 2 units to the left along the x-axis and 1 unit down along the y-axis.

Once we have all the points plotted, we can draw a line through them to represent the equation graphically. If plotted correctly, these points should form a straight line, demonstrating that we have correctly graphed the linear equation. Remember, ensuring accuracy while plotting is crucial as even small errors in placement can significantly affect the graph's shape.