Problem 34 - Introductory Algebra for College Students

Question

Solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l}x-y=2 \\ 3 x-3 y=-6\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system is \(\{\}\), meaning no solution exists.

Step 1 - Rewrite the equations in slope-intercept form

To graph the equations easily, rewrite them in the form \(y=mx+b\), where \(m\) is the slope and \(b\) is the y-intercept. The first equation becomes \(y=x-2\) and the second equation, after dividing by 3, becomes \(y=x+2\).

Step 2 - Graph the equations

On the same pair of axes, graph each equation. Each is a straight line with a slope of 1, but they have different y-intercepts. The line \(y=x-2\) crosses the y-axis at -2 and the line \(y=x+2\) crosses the y-axis at 2.

Step 3 - Find the intersection

Looking at the graph, we see that the lines do not intersect. This means there is no solution to the system of equations, as no pair of (x, y) values satisfies both equations at the same time.

Step 4 - Express the solution in set notation

Since the graph shows no intersection, there is no solution to this system, which is written in set notation as \(\{\}\), the empty set.

Key Concepts

Graphing Method

The graphing method is a visual way to solve a system of equations. This method involves plotting each equation on the same graph and observing where the lines intersect. It can be a handy tool for understanding and solving linear systems.

When you graph the lines, you're essentially mapping out all the possible solutions for each equation. The solution to the system is the point where the lines cross, also known as the point of intersection. If the lines intersect at a single point, that point is the one solution that satisfies both equations.

• If the lines are parallel, like in our exercise, they never intersect, meaning there is no common solution.
• If the lines coincide or overlap perfectly, there are infinitely many solutions.

Consider graphing an introductory step. It provides a clear picture that makes it easier to see the relationships between different equations.

Slope-Intercept Form

In order to graph an equation easily, converting it into the slope-intercept form is key. The slope-intercept form is given by: \[ y = mx + b \], where:

• \( m \) represents the slope of the line, indicating how steep the line is.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

For the initial system of equations, each can be rewritten in this form to facilitate graphing. For example, converting the equations from the exercise:

• From \( x - y = 2 \), we rearrange to \( y = x - 2 \).
• From \( 3x - 3y = -6 \), divide by 3 to get \( y = x + 2 \).

These equations in slope-intercept form give an immediate view of each line's slope and intercept, making plotting straightforward.

Intersection of Lines

The intersection of lines refers to the point where two lines cross each other on a graph. This point is significant because it represents the solution to the system of equations - the values of \( x \) and \( y \) that satisfy both equations simultaneously.

In our given problem, the lines

• \( y = x - 2 \)
• \( y = x + 2 \)

do not intersect because they are parallel. To confirm parallel lines, check both their slopes. Lines with the same slope but different y-intercepts will never meet.

Understanding when lines intersect or are parallel helps you determine whether a system of equations has one solution, infinitely many solutions, or no solution.

Solution Set

A solution set is a term used to describe the set of all values that solve a given mathematical problem. In the context of systems of equations, the solution set consists of all points \( (x, y) \) shared by the equations.

For our exercise, since the two lines are parallel and do not intersect, there is no point \( (x, y) \) satisfying both equations. Thus, the solution set is empty. Mathematically, this is denoted as \( \{\} \), the empty set. If the lines had intersected at one point, the solution set would include just that single point. Similarly, if the lines were coincident, they would share many points, leading to an infinite solution set.