Problem 52 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the lines in the system. Use the graphs to determine whether the system is consistent or inconsistent. If the system is consistent, determine the solution. Verify your results algebraically. $$\underline{\phantom{xxx}}\left\\{\begin{array}{l} -6 x+4 y=-9 \\ 4.5 x-3 y=6.75 \end{array}\right.$$

Step-by-Step Solution

Answer

The system is inconsistent and has no solution.

Step 1: Graph the Equations

To graph the given equations, -6x + 4y = -9 and 4.5x - 3y = 6.75, rewrite them in the slope-intercept form, \(y = mx + b\), where m is the slope and b is the y-intercept. The first equation, simplified, gives: \(y = 1.5x + 2.25\). The second equation simplified gives: \(y = 1.5x - 2.25\).

Step 2: Analyze the Graphs

By graphing these two lines, it can be seen that the lines are parallel, which means there is no point of intersection. Two parallel lines never intersect and therefore, the system of equations is inconsistent.

Step 3: Verify Results Algebraically

To verify the graphical conclusion algebraically, try to solve the system of equations. By comparing the coefficients of the variables in both equations, you can see that the coefficients of x and y in the first equation are -2 times the corresponding coefficients in the second equation, whereas the constants on the right do not have the same ratio. Therefore, algebraically, we can verify that the system is inconsistent as there is no solution that can satisfy both equations.

Key Concepts

Consistent and Inconsistent Systems

When working with systems of equations, we are essentially looking for the set of values that satisfy all equations simultaneously. A system is called consistent if there is at least one set of values (called the solution) that works for all equations. Conversely, a system is inconsistent if there is no possible solution that satisfies all the equations at the same time. When we graph consistent systems, the lines will intersect at the point that represents the solution to the system. On the other hand, inconsistent systems are graphed as parallel lines, which never meet and hence do not have a point of intersection or a shared solution. Understanding whether a system is consistent or inconsistent is crucial, as it determines whether we can find a solution or conclude that no solution exists.

For example, in the exercise provided, after graphing the equations on a utility graph, we can observe that the lines are distinct and parallel, which indicates that there is no intersection point. It directly suggests that the system is inconsistent, and there is no common solution to these lines, a fact that can also be confirmed through algebraic verification.

Slope-Intercept Form

The slope-intercept form of a linear equation is an incredibly useful tool in graphing systems of equations. It is written as y = mx + b, where m represents the slope of the line and b represents the y-intercept, which is the point where the line crosses the y-axis. This form makes it easy to graph a line because it gives a clear starting point (the y-intercept) and indicates the direction and steepness of the line (the slope).

In the exercise, converting the given equations into the slope-intercept form enabled us to graph them easily and identify the slope and y-intercept. Both lines in the exercise had the same slope but different y-intercepts, putting them in a position that they will never cross each other—hence, indicating an inconsistent system. Using the slope-intercept form is often the first step in graphically solving a system of equations, as it lays out the path of each individual line.

Parallel Lines

Parallel lines have a lot of significance in the context of solving systems of equations. These are lines that run alongside each other and have the same slope, indicated by the m value in the slope-intercept form of the equation, but they never intersect regardless of how far they are extended. This indicates that parallel lines share no common points. Graphically, parallel lines can be easily identified by comparing their slopes and noticing the equal separation distance along their entire length.

When we graph a system of equations and find that the lines are parallel, as in the exercise provided, we have found a visual representation of an inconsistent system. The similarity in the slopes of both equations ( y = 1.5x + 2.25 and y = 1.5x - 2.25) proves the lines are parallel and thus will never meet.

Algebraic Verification

Algebraic verification is a rigorous way of proving graphically obtained results. It involves manipulating the equations in a system to check if they can yield a consistent solution. This is often done by solving the equations simultaneously through methods like substitution or elimination. If these algebraic methods lead to a contradiction or an impossibility, such as '0 = 1', then we can confirm that the system of equations is inconsistent.

In the provided exercise, the algebraic verification process involves observing that the coefficients of the equations are multiples of each other but the constants are not. This is a clear indication that the system has no solution, reflecting what we observed graphically - the parallel lines. Algebraic verification serves as a foundational check, ensuring the accuracy of our graphical interpretations.