Problem 78 - 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. $$y=x-\frac{1}{2}$$

Step-by-Step Solution

Answer

Plotting the points in the solutions (-2,-3/2), (-1,-1.5), (0,-0.5), (1,0.5), and (2,1.5) on a graph gives us the graph of the linear equation \(y=x-1/2\).

Step 1: Understand the equation

Here, we are given the linear equation \(y=x-1/2\). This is a linear equation in slope-intercept form, where 1 is the slope, and \(-1/2\) is the y-intercept. This means that the line crosses the y-axis at \(-1/2\) and rises 1 unit for every 1 unit it moves to the right along the x-axis.

Step 2: Construct a table of values

Since we're looking for five solutions, let's choose five different values for \(x\) and then solve for the corresponding \(y\) values using the equation. Let's use \(x\) values of -2, -1, 0, 1, and 2.

Step 3: Find corresponding y-values

Now, we substitute each \(x\) value into the equation \(y=x-1/2\) to solve for the corresponding \(y\) values. For example, if \(x=-2\), we substitute -2 into the equation: \(y=(-2)-1/2=-3/2\). So, \(-2,-3/2\) is one solution. Following this method, the solutions are: (-2,-3/2), (-1,-1.5), (0,-0.5), (1,0.5), and (2,1.5).

Step 4: Plot the points and draw the line

Now that we have five solutions, we can plot these points on a graph. After we've plotted the points, we can draw a straight line that passes through all five points. This line represents the set of all solutions to our equation.

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the easiest ways to write linear equations. In this form, a linear equation looks like this: \[ y = mx + b \]where \( m \) is the slope, and \( b \) is the y-intercept.
The slope \( m \) tells us how steep the line is and in which direction it goes:

• A positive slope means the line goes up as you move to the right.
• A negative slope means the line goes down as you move to the right.

The y-intercept \( b \) is the point where the line crosses the y-axis. This means when \( x = 0 \), \( y = b \).
In the equation \( y = x - \frac{1}{2} \), the slope \( m \) is 1. This indicates that for every increase of 1 unit in the \( x \)-direction, the \( y \)-value increases by 1 unit. The y-intercept here is \(-\frac{1}{2}\), which means the line crosses the y-axis at \(-0.5\). This form is incredibly useful for graphing and quickly understanding the behavior of the line.

Graphing Equations

Graphing linear equations shows what solutions look like visually. Once we understand the slope-intercept form, plotting the graph becomes straightforward.
After determining the slope \( m \) and the y-intercept \( b \):

• Start by plotting the y-intercept on the graph. For our equation, plot the point \( (0, -0.5) \).
• Use the slope to find another point. With a slope of 1, start from the y-intercept and move up 1 unit and to the right 1 unit, placing another point at \( (1, 0.5) \).
• Continue marking additional points using the slope for accuracy.

For multiple linear equations, each line represents different solutions. Graphing helps us visually verify solutions and understand relationships between variables.

Table of Values

Creating a table of values is a powerful method to find specific points on a graph. It's especially helpful when dealing with equations like our linear one.
To make a table of values:

• Select several \( x \) values. For this exercise, we chose \(-2, -1, 0, 1, \) and \(2\).
• Insert each \( x \) value into the equation to solve for \( y \).
• List these \( (x, y) \) pairs in a table format. For instance, with \( x = -2 \), substitute into \( y = x - \frac{1}{2} \) to get \( y = -\frac{3}{2} \).

This approach finds exact points, which can then be plotted on a graph.
The generated points alone can draft the line if drawn accurately. This method reinforces understanding of substitutions and the linear relationship between \( x \) and \( y \).