Solving Systems Of Equations A Deep Dive Into Infinite Solutions
This article delves into the fascinating world of systems of equations, specifically addressing the scenario presented:
-5.9x - 3.7y = -2.1
5.9x + 3.7y = 2.1
We will explore various methods to solve this system and ultimately demonstrate why the correct answer is C. infinitely many solutions. Understanding why this is the case requires a solid grasp of the underlying principles of linear equations and their graphical representations.
Understanding Systems of Equations
In the realm of mathematics, a system of equations is a set of two or more equations containing the same variables. The solutions to a system of equations are the values for the variables that make all equations in the system true simultaneously. Graphically, each equation in a system represents a line (for linear equations), and the solution(s) to the system correspond to the point(s) where the lines intersect. There are three primary possibilities for the solutions of a system of two linear equations:
- One Unique Solution: The lines intersect at a single point.
- No Solution: The lines are parallel and never intersect.
- Infinitely Many Solutions: The lines are coincident, meaning they overlap completely.
Methods for Solving Systems of Equations
Several methods exist for solving systems of equations, each with its strengths and weaknesses. We will explore two common methods – elimination and substitution – to demonstrate how they apply to the given system.
1. Elimination Method
The elimination method focuses on manipulating the equations in the system to eliminate one variable, allowing us to solve for the other. This method is particularly effective when the coefficients of one variable are opposites or easily made opposites. Let's apply this method to our system:
-5.9x - 3.7y = -2.1
5.9x + 3.7y = 2.1
Notice that the coefficients of x (-5.9 and 5.9) are opposites. Similarly, the coefficients of y (-3.7 and 3.7) are also opposites. This sets the stage perfectly for elimination. If we add the two equations together, we get:
(-5.9x - 3.7y) + (5.9x + 3.7y) = -2.1 + 2.1
Simplifying this, we find:
0 = 0
This result, 0 = 0, is a true statement, but it doesn't give us specific values for x or y. This is a crucial indicator that the system has infinitely many solutions. The equations are essentially representing the same line. When we get an identity like 0=0, it signifies that the two equations are dependent and represent the same line.
2. Substitution Method
The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This method is useful when one equation can be easily solved for one variable in terms of the other. Let's attempt to use this method with our system. First, let's solve the second equation for x:
5. 9x + 3.7y = 2.1
6. 9x = 2.1 - 3.7y
7. = (2.1 - 3.7y) / 5.9
Now, substitute this expression for x into the first equation:
-5.9 * ((2.1 - 3.7y) / 5.9) - 3.7y = -2.1
Simplifying this equation:
-(2.1 - 3.7y) - 3.7y = -2.1
-2.1 + 3.7y - 3.7y = -2.1
-2.1 = -2.1
Again, we arrive at a true statement (-2.1 = -2.1), but no specific solution for y. This further confirms that the system has infinitely many solutions. Just like the elimination method, the substitution method reveals that we are essentially dealing with the same equation expressed in two slightly different forms.
Graphical Interpretation
To solidify our understanding, let's consider the graphical interpretation of this system of equations. Each equation represents a line in the coordinate plane. If we rewrite both equations in slope-intercept form (y = mx + b), we can clearly see their relationship:
First Equation:
-5.9x - 3.7y = -2.1
-3.7y = 5.9x - 2.1
y = (-5.9/3.7)x + (2.1/3.7)
Second Equation:
5. 9x + 3.7y = 2.1
6. 7y = -5.9x + 2.1
y = (-5.9/3.7)x + (2.1/3.7)
Notice that both equations have the same slope (-5.9/3.7) and the same y-intercept (2.1/3.7). This means that the two equations represent the exact same line. When two lines are identical, they overlap completely, and every point on the line is a solution to both equations. Therefore, there are infinitely many solutions.
Why Not the Other Options?
Let's briefly address why the other options are incorrect:
- A. (0, -2.1) and B. (0, 2.1): These are specific coordinate pairs. While they might satisfy one of the equations, they need to satisfy both to be a solution to the system. Since we have infinitely many solutions, a single point is not sufficient.
- D. No Solution: This would be the case if the lines were parallel but had different y-intercepts. In that scenario, they would never intersect. However, as we've demonstrated, our lines are the same line, not parallel.
The Significance of Infinite Solutions
Systems with infinitely many solutions often arise in various mathematical and real-world contexts. They indicate a dependency between the equations, meaning one equation can be derived from the other. This can represent situations where constraints are redundant or where there is a range of possibilities that satisfy the given conditions. For example, in economics, a system of equations representing supply and demand might have infinitely many solutions under certain market conditions, indicating a flexible equilibrium.
Conclusion
The system of equations:
-5.9x - 3.7y = -2.1
5.9x + 3.7y = 2.1
has infinitely many solutions (Option C). This conclusion is reached through both algebraic methods (elimination and substitution) and graphical analysis. The key takeaway is that the two equations represent the same line, leading to an infinite number of points satisfying both equations. Understanding this concept is crucial for solving various mathematical problems and interpreting real-world scenarios modeled by systems of equations. The ability to recognize dependent equations and systems with infinite solutions is a valuable skill in mathematics and related fields. This detailed exploration highlights the importance of not just finding a solution, but also understanding the nature of the solution set and the relationships between the equations within a system.
