Problem 42 - College Algebra with Modeling and Visualization

Question

Graph each system of equations and find any solutions. Check your answers. Identify the system as consistent or inconsistent. If the system is consistent, state whether the equations are dependent or independent. $$ \begin{aligned} 3 x-y &=-2 \\ -3 x+y &=2 \end{aligned} $$

Step-by-Step Solution

Answer

The system is consistent and the equations are dependent with infinitely many solutions.

Step 1: Write Each Equation in Slope-Intercept Form

To write the given equations in the slope-intercept form, which is \(y = mx + b\), solve each equation for \(y\). For the first equation, \(3x - y = -2\), solve for \(y\):\\(3x - y = -2\) becomes \(y = 3x + 2\).For the second equation, \(-3x + y = 2\), solve for \(y\):\\(-3x + y = 2\) becomes \(y = 3x + 2\).

Step 2: Graph the Equations

Graph the two equations \(y = 3x + 2\) and \(y = 3x + 2\) on the same coordinate axes. Both lines have the same slope (\(m=3\)) and y-intercept (\(b=2\)). Since they are the same line, the graphs will overlap completely.

Step 3: Identify Number of Solutions and Consistency

Since both equations represent the same line, any point on the line is a solution. This means there are infinitely many solutions. Therefore, the system is consistent because at least one solution exists.

Step 4: Determine Equation Dependency

Since both equations are essentially multiples of each other (they are equivalent), they are dependent. Dependent equations yield the same graph and thus, do not provide additional information.

Step 5: Verify the Solution

Since both equations have been written correctly and graphically they result in the same line, verification confirms that any solution for one will satisfy the other, proving our conclusion.

Key Concepts

Consistent vs Inconsistent Systems

When dealing with a system of equations, it’s crucial to determine if the system has solutions, meaning if it's "consistent," or if it lacks solutions, making it "inconsistent." A consistent system has at least one solution. When you solve for the variables, they equal the same values in both equations. An inconsistent system, on the other hand, means there are no possible values for the variables that can satisfy all of the equations simultaneously. In graphical terms, consistent systems will have lines that intersect at least at one point or overlap completely.
In the original exercise, because both equations yield the same line, it is a consistent system, demonstrating an infinite number of solutions.

Dependent vs Independent Equations

Once we've established that a system is consistent because it has solutions, we next determine if these equations are dependent or independent. Independent equations will intersect at only one point. This means their slopes are different, indicating different lines intersecting at one point on a graph. Dependent equations, however, refer to multiple equations representing the same line, hence providing no new information about the system.
In our given exercise, since the equations both convert to the equation \(y = 3x + 2\), they describe the same line. Therefore, these equations are dependent and graphically overlap entirely.

Slope-Intercept Form

The slope-intercept form is a standard way of writing linear equations, which is \(y = mx + b\). In this equation, \(m\) represents the slope or steepness of the line, while \(b\) is the y-intercept, or where the line crosses the y-axis. Converting equations to this form is helpful for graphing and comparing the lines in a system easily.
For the first equation, solving \(3x - y = -2\) gives \(y = 3x + 2\). Similarly, solving \(-3x + y = 2\) also gives \(y = 3x + 2\). With both in this form, it's clear they have the same slope of 3 and intercept at 2, confirming they are the same line.

Graphical Solutions

Graphing systems of equations is a visual method to find solutions. Each equation is represented as a line on a coordinate plane, and where they intersect is the solution. If the lines meet at a single point, it’s a unique solution, meaning the equations are independent. If lines are parallel and never meet, there are no solutions, indicating an inconsistent system.

In scenarios where lines completely overlap, as with the given system \(y = 3x + 2\) for both equations, the solution is all points along that line, indicating infinitely many solutions. This visual confirmation shows our system is consistent and dependent, since both equations form identical overlapping lines.