Solving Systems Of Equations Determining The Correct Solution
Understanding Systems of Equations
Systems of equations are fundamental in mathematics and find applications in various real-world scenarios. These systems involve two or more equations with the same set of variables. Solving a system of equations means finding the values of the variables that satisfy all equations simultaneously. In simpler terms, it's like finding the common ground where all the equations agree. The solution to a system of equations represents the point(s) where the lines or curves represented by the equations intersect. The nature of these intersections determines the type of solution the system has – no solution, a unique solution, or infinitely many solutions.
To solve the given system, we can employ several methods, including substitution, elimination, and graphical methods. Each method has its strengths and is suitable for different types of systems. The substitution method involves solving one equation for one variable and substituting that expression into the other equation. This reduces the system to a single equation with one variable, which can then be easily solved. The elimination method, on the other hand, involves manipulating the equations to eliminate one of the variables, again resulting in a single equation with one variable. The graphical method involves plotting the equations on a coordinate plane and finding the point(s) of intersection, which represent the solution(s) to the system. Understanding these methods is crucial for tackling various mathematical problems and real-world applications involving systems of equations.
Systems of equations are more than just abstract mathematical concepts; they are powerful tools for modeling and solving real-world problems. From determining the optimal mix of products in manufacturing to predicting the trajectory of a projectile in physics, systems of equations provide a framework for analyzing complex relationships between variables. In economics, they can be used to model supply and demand curves, finding equilibrium prices and quantities. In engineering, they are essential for designing structures, circuits, and systems that meet specific criteria. Moreover, systems of equations play a vital role in computer science, particularly in areas such as computer graphics, optimization, and machine learning. The ability to solve systems of equations efficiently and accurately is a valuable skill in many professions, highlighting the practical significance of this mathematical concept.
Analyzing the Given System of Equations
The given system of equations is:
3x + y = 17
x + 2y = 49
This system consists of two linear equations with two variables, x and y. To determine the solution, we need to find the values of x and y that satisfy both equations simultaneously. We can use either the substitution method or the elimination method to solve this system. Let's start by exploring the substitution method. We can solve the first equation for y:
y = 17 - 3x
Now, substitute this expression for y into the second equation:
x + 2(17 - 3x) = 49
Simplifying this equation will give us the value of x. Once we have the value of x, we can substitute it back into either of the original equations or the expression for y to find the value of y. Alternatively, we can use the elimination method. To do this, we can multiply the first equation by -2 to eliminate the y variable when we add the equations together. This will give us a single equation in terms of x, which we can solve. After finding x, we can substitute it back into one of the original equations to find y. Both methods are equally valid and will lead to the same solution. The choice of method often depends on personal preference or which method seems easier for a particular system of equations.
By carefully applying these methods, we can determine whether the system has a unique solution, no solution, or infinitely many solutions. A unique solution means the lines intersect at exactly one point. No solution means the lines are parallel and never intersect. Infinitely many solutions means the lines are coincident, meaning they are the same line. This analysis is crucial not only for solving the system but also for understanding the geometric relationship between the lines represented by the equations.
Understanding the nature of solutions is a core concept in linear algebra. When we solve a system of linear equations, we are essentially trying to find the intersection points of the lines (in two dimensions) or planes (in three dimensions) represented by these equations. A unique solution corresponds to a single intersection point, indicating that the lines or planes meet at a specific location in space. No solution implies that the lines or planes are parallel and never intersect, meaning there is no common point that satisfies all equations simultaneously. Infinitely many solutions occur when the lines or planes overlap, indicating that every point on the line or plane satisfies all equations. These concepts extend to systems with more variables and equations, providing a foundation for solving complex problems in various fields.
Solving the System Using Elimination Method
Let's proceed with the elimination method to solve the system:
3x + y = 17
x + 2y = 49
To eliminate y, we can multiply the first equation by -2:
-2(3x + y) = -2(17)
-6x - 2y = -34
Now we have the modified system:
-6x - 2y = -34
x + 2y = 49
Add the two equations:
(-6x - 2y) + (x + 2y) = -34 + 49
-5x = 15
Divide by -5 to solve for x:
x = -3
Now that we have the value of x, we can substitute it into either of the original equations to find y. Let's use the first equation:
3x + y = 17
3(-3) + y = 17
-9 + y = 17
Add 9 to both sides to solve for y:
y = 26
So, the solution to the system is x = -3 and y = 26. This means that the point (-3, 26) is the intersection of the two lines represented by the equations. This unique solution indicates that the system is consistent and the lines are independent, meaning they are not parallel or coincident.
The elimination method is a powerful technique for solving systems of linear equations because it systematically eliminates one variable at a time, simplifying the problem. By multiplying one or both equations by appropriate constants, we can make the coefficients of one variable opposites. When we add the equations together, this variable cancels out, leaving us with a single equation in one variable. This equation can then be easily solved. The solution for this variable is then substituted back into one of the original equations to solve for the other variable. This method is particularly useful when the equations are in standard form (Ax + By = C) and the coefficients are such that elimination can be achieved with simple multiplication. The elimination method is a cornerstone of linear algebra and is used extensively in various applications, including network analysis, circuit design, and economic modeling.
Determining the Nature of the Solution
We found that the solution to the system is x = -3 and y = 26. This means the system has a single, unique solution. Geometrically, this corresponds to the two lines intersecting at one point, (-3, 26), on the coordinate plane. A single solution indicates that the two equations are independent and consistent. Independent means that the equations represent distinct lines, and consistent means that the lines intersect at least once.
If the system had no solution, the lines would be parallel and never intersect. This would occur if the equations had the same slope but different y-intercepts. In this case, any attempt to solve the system would lead to a contradiction, such as 0 = a non-zero number. If the system had infinite solutions, the two equations would represent the same line. This would occur if one equation is a multiple of the other. In this case, solving the system would result in an identity, such as 0 = 0, indicating that any point on the line satisfies both equations.
Understanding the different types of solutions is essential for interpreting the results of solving systems of equations. A unique solution is the most straightforward case, indicating a clear and specific answer. No solution highlights an inconsistency in the equations, suggesting that the model they represent may need to be re-evaluated. Infinitely many solutions suggest a redundancy in the equations, indicating that some information may be unnecessary or that the model has degrees of freedom. Recognizing these different scenarios allows for a deeper understanding of the relationships between variables and the systems they form. This knowledge is invaluable in various fields, including engineering, economics, and computer science, where systems of equations are used to model and solve complex problems.
Conclusion
The statement that best describes the solution to the given system of equations is:
C. It has a single solution: x = -3, y = 26
This conclusion is reached by solving the system using the elimination method and finding a unique pair of values for x and y that satisfy both equations. The understanding of different types of solutions – unique, none, or infinite – is crucial in solving systems of equations and interpreting their geometric representation.
In summary, the ability to solve systems of equations is a fundamental skill in mathematics with wide-ranging applications. Whether using substitution, elimination, or graphical methods, the key is to understand the underlying concepts and choose the most efficient approach for a given system. The nature of the solution – unique, none, or infinite – provides valuable insights into the relationships between variables and the models they represent. Mastering these concepts is essential for success in mathematics and various STEM fields, where systems of equations are used to model and solve real-world problems.
