Problem 62 - 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. $$y=2 x-1$$

Step-by-Step Solution

Answer

Five solutions for the equation \(y=2x-1\) include (-2, -5), (-1, -3), (0, -1), (1, 1), and (2, 3). The graph of the equation is a straight line passing through these points.

Step 1: Find the solutions

To find the solutions for the equation \(y=2x-1\), substitute different values for \(x\). For instance, when \(x=-2\), \(x=-1\), \(x=0\), \(x=1\), and \(x=2\), compute the corresponding \(y\) values using the given equation.

Step 2: Create a table of values

Once the solutions are found, list them in a table format, each \(x\) value corresponding to its \(y\) value. The table should look like: \n\n \[\begin{tabular}{|c|c|}\hlinex & y \\hline-2 & -5 \-1 & -3 \0 & -1 \1 & 1 \2 & 3 \\hline\end{tabular}\]

Step 3: Plot the solutions

Using the table of values, plot each solution as a point on the graph, where the \(x\) value is on the horizontal axis and the \(y\) value is on the vertical axis. Once all points are plotted, draw a straight line that pass through all points. This line is the graph of the equation \(y=2x-1\).

Key Concepts

Graphing Linear Equations

When working with linear equations, graphing is a powerful way to visualize their solutions. A linear equation in two variables, such as \( y = 2x - 1 \), forms a straight line when plotted on a coordinate plane. The "linear" in linear equations means the graph will always be a straight line. The general form of a linear equation is \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants. However, many times it’s seen in the slope-intercept form \( y = mx + b \), where \( m \) is the slope of the line, and \( b \) is the y-intercept.

To graph a linear equation, you follow these steps:

• Find the solutions or points that satisfy the equation, often done by choosing values for \( x \).
• Create a table of these values, pairing each \( x \) with its corresponding \( y \).
• Plot these points on the graph and draw a line through them.

The slope \( m = 2 \) in this equation suggests that for every one unit increase in \( x \), \( y \) increases by two units. The y-intercept \( -1 \) is the point where the line crosses the y-axis.
By understanding these fundamental properties, graphing any linear equation becomes straightforward and systematic.

Solution Sets

A solution set is a collection of ordered pairs \((x, y)\) that satisfy a given linear equation. For \( y = 2x - 1 \), substituting different values for \( x \) provides different values for \( y \). Each of these pairs \((x, y)\) represents a solution to the equation.

Typically, to start finding the solution set:

• Select a range of \( x \) values, covering both negative and positive numbers can help in understanding the behavior of the line.
• Substitute these \( x \) values into the equation to solve for the corresponding \( y \).
• List these \( (x, y) \) pairs. These points make up the solution set.

This approach doesn't just apply to finding particular solution sets, but also helps in confirming that a line indeed represents all solutions to the equation. Any point on the line, infinite points actually, satisfies the equation, making the line the geometric representation of the entire solution set.

Table of Values

Constructing a table of values is a practical step for graphing equations. It helps organize your solution set clearly and systematically. For example, for the equation \( y = 2x - 1 \), we could choose different \( x \) values: \(-2, -1, 0, 1, 2\).

Once the \( x \) values are chosen, find the associated \( y \) values:

• For \( x = -2 \), \( y = 2(-2) - 1 = -5 \).
• For \( x = -1 \), \( y = 2(-1) - 1 = -3 \).
• For \( x = 0 \), \( y = 2(0) - 1 = -1 \).
• And so on for the remaining values.

This process assists in creating a table of values:

• \((-2, -5)\)
• \((-1, -3)\)
• \((0, -1)\)
• \((1, 1)\)
• \((2, 3)\)

With these ordered pairs, you can plot each point on a coordinate plane. The series of points will align, allowing you to draw the line representing the equation. This systematic approach highlights the relationship between \( x \) and \( y \) values in the equation, visualizing their connection on the graph perfectly.