Problem 80 - 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. $$y=3, \text { or } y=0 x+3$$

Step-by-Step Solution

Answer

The five solutions/pairs are (-2,3), (-1,3), (0,3), (1,3), and (2,3). The graph of both equations is a horizontal line crossing the y-axis at the point (0,3).

Step 1: Identify the type of lines

The given equations \(y=3\) and \(y=0x+3\) are forms of linear equations. In the equation \(y=3\), it is a horizontal line crossing the y-axis at (0,3). The equation \(y=0x+3\) is in the slope-intercept form where 0 is the slope (m), meaning it's a horizontal line and 3 is the y-intercept (b), where the line intersects with the y-axis.

Step 2: Construct a table of values for the equation

To find at least five solutions, choose five values for x and substitute them into the equation. As the equation is the same in other forms, y-values will always be 3. So, the five pairs can be (-2,3), (-1,3), (0,3), (1,3), and (2,3). These are points on the line.

Step 3: Plotting the points and drawing the graph

On a graph paper, locate these points and join them. You will notice that all these points align vertically on y=3.

Key Concepts

Linear Equations in Two Variables

Linear equations in two variables are foundational to understanding algebra and represent geometrically as lines on a coordinate plane. An equation like \(y = mx + b\) is a standard representation where \(x\) and \(y\) are variables, \(m\) indicates the slope of the line, and \(b\) represents the y-intercept, the point where the line crosses the y-axis.

When the equation is simplified, such as \(y = 3\), it describes a special case where the line is horizontal, indicating that no matter the value of \(x\), the value of \(y\) remains constant. It's straightforward in plotting since for any \(x\), the pair \( (x, 3)\) is a solution, showcasing the interplay between the algebraic equation and its graphical representation.

Table of Values

A table of values is a tool used to determine a set of points that satisfy a given equation. It's particularly useful for visualizing how changes in one variable affect the other.

For the equation \(y = 3\), the table of values is simple—the value of \(y\) is always 3, and \(x\) can be any real number. To graph the linear equation, you typically select a range of \(x\) values and compute the corresponding \(y\) values. With the equation \(y = 0x + 3\), it's clear that the slope is 0, which means for any \(x\), \(y\) will again be 3. In this case, our table can include pairs like \( (-2, 3), (-1, 3), (0, 3), (1, 3),\) and \( (2, 3)\) as pointed out in the exercise solution, evidencing the constant nature of the y-value.

Slope-Intercept Form

The slope-intercept form of a linear equation is given by \(y = mx + b\), an instructive format to graph lines quickly. The \(m\) stands for the slope, determining the steepness and direction of the line, while \(b\) is the y-intercept, where the line crosses the y-axis.

In our exercise, \(y = 0x + 3\) can be seen as \(y = 3\) since multiplying \(x\) by zero yields zero, effectively simplifying to the constant equation. This equation indicates a line with a slope of zero, which is a horizontal line. Horizontal lines are unique in that their slope is zero hence no vertical change as \(x\) varies. The y-intercept is 3, meaning this line crosses the y-axis at the point \( (0,3)\). Students can connect these ideas to plot a precise graph by starting at the y-intercept and recognizing that the line will remain at that y-value.