Solving Systems Of Equations A Comprehensive Guide
The question at hand involves finding the solution to a system of two linear equations:
y = 1.5x - 4
y = -x
To solve this system, we need to find the values of x and y that satisfy both equations simultaneously. This can be achieved through several methods, including substitution, elimination, and graphing. Each method provides a unique approach to arrive at the correct solution. In this comprehensive guide, we will explore the substitution method in detail, offering a step-by-step explanation to ensure clarity and understanding. Furthermore, we will delve into the practical implications of solving systems of equations, highlighting real-world scenarios where these mathematical skills prove invaluable. Understanding how to solve systems of equations is not just an academic exercise; it is a fundamental skill that enhances problem-solving abilities across various domains.
Solving the System of Equations
Using the Substitution Method
The substitution method is a powerful technique for solving systems of equations. It involves solving one equation for one variable and then substituting that expression into the other equation. This process reduces the system to a single equation with one variable, making it easier to solve. Let's apply this method to our system:
y = 1.5x - 4
y = -x
Since both equations are already solved for y, we can substitute the expression for y from the second equation into the first equation. This gives us:
-x = 1.5x - 4
Now we have a single equation with one variable, x. To solve for x, we need to isolate it on one side of the equation. Let's add x to both sides:
-x + x = 1.5x + x - 4
0 = 2.5x - 4
Next, add 4 to both sides:
4 = 2.5x
Finally, divide both sides by 2.5 to solve for x:
x = 4 / 2.5
x = 1.6
Now that we have the value of x, we can substitute it back into either of the original equations to find the value of y. Let's use the second equation, y = -x:
y = -1.6
Thus, the solution to the system of equations is x = 1.6 and y = -1.6. This means the point where the two lines intersect on a graph is (1.6, -1.6). The substitution method simplifies the process by reducing a two-variable problem into a single-variable one, making it an efficient tool for solving systems of equations. This approach is particularly useful when one of the equations is already solved for one variable or can be easily manipulated to do so. Furthermore, understanding the substitution method enhances one's ability to tackle more complex mathematical problems involving multiple variables and equations.
Verifying the Solution
To ensure our solution is correct, it's essential to verify it by substituting the values of x and y back into both original equations. This step confirms that the solution satisfies both equations simultaneously, which is a fundamental requirement for any solution to a system of equations. Let's start with the first equation:
y = 1.5x - 4
Substitute x = 1.6 and y = -1.6:
-1.6 = 1.5(1.6) - 4
-1.6 = 2.4 - 4
-1.6 = -1.6
The equation holds true. Now, let's verify the second equation:
y = -x
Substitute x = 1.6 and y = -1.6:
-1.6 = -1.6
This equation also holds true. Since the values of x and y satisfy both equations, we can confidently conclude that our solution (1.6, -1.6) is correct. Verifying the solution is a crucial step in problem-solving, as it helps to identify any errors made during the calculation process. This practice not only ensures accuracy but also reinforces the understanding of the underlying mathematical concepts. In the context of systems of equations, verification solidifies the understanding that the solution represents the point of intersection of the lines represented by the equations.
Graphical Interpretation
Solving a system of equations graphically involves plotting the equations on a coordinate plane and identifying the point where the lines intersect. This intersection point represents the solution to the system, as it is the only point that satisfies both equations simultaneously. In our case, we have the following equations:
y = 1.5x - 4
y = -x
Each of these equations represents a straight line. The first equation, y = 1.5x - 4, has a slope of 1.5 and a y-intercept of -4. The second equation, y = -x, has a slope of -1 and a y-intercept of 0. When we plot these lines on a graph, we can visually determine their intersection point. The intersection point will have coordinates (x, y) that satisfy both equations. By graphing these lines, we would observe that they intersect at the point (1.6, -1.6), which confirms our algebraic solution. Graphical interpretation provides a visual understanding of the solution to a system of equations. It helps to see the relationship between the equations and their solution in a tangible way. This method is particularly useful for students who are visual learners, as it connects the abstract concept of solving equations with a concrete visual representation. Furthermore, graphical solutions can be used to estimate solutions for more complex systems of equations or inequalities where algebraic methods may be more challenging to apply.
Choosing the Correct Answer
Given the options:
A. (-1.6, 1.6) B. (-1.5, 1.5) C. (1.5, -1.5) D. (1.6, -1.6)
Based on our calculations, the correct solution is (1.6, -1.6), which corresponds to option D. The systematic approach we followed, including the substitution method and verification, ensures that we have arrived at the accurate answer. When presented with multiple-choice questions, it's crucial to have a clear understanding of the problem-solving process. By applying a structured method and verifying the solution, you can confidently choose the correct answer and avoid common pitfalls. In the context of solving systems of equations, understanding the different methods available, such as substitution, elimination, and graphing, allows you to select the most efficient approach for a given problem. Additionally, the ability to interpret the solution graphically provides a valuable check on the algebraic solution, further enhancing your confidence in the answer.
Real-World Applications
Solving systems of equations is not merely a theoretical exercise confined to the classroom; it has numerous practical applications in various real-world scenarios. These applications span across different fields, including economics, engineering, physics, and computer science, highlighting the versatility and importance of this mathematical skill. Let's explore some specific examples:
Economics
In economics, systems of equations are used to model supply and demand curves. The equilibrium point, where the supply and demand curves intersect, represents the market price and quantity. Economists use systems of equations to analyze market trends, predict price changes, and make informed decisions about resource allocation. For instance, consider a scenario where the supply equation is given by y = 2x + 10 and the demand equation is given by y = -x + 40, where y represents the price and x represents the quantity. Solving this system of equations will yield the equilibrium price and quantity, providing valuable insights into the market dynamics.
Engineering
Engineers use systems of equations to design structures, circuits, and systems. For example, in structural engineering, systems of equations are used to analyze the forces acting on a structure and ensure its stability. In electrical engineering, systems of equations are used to analyze circuits and determine the current and voltage in different components. Consider a simple circuit with two loops and two voltage sources. Applying Kirchhoff's laws will result in a system of linear equations that can be solved to determine the currents flowing through each loop. This application underscores the critical role of systems of equations in ensuring the safety and efficiency of engineered systems.
Physics
In physics, systems of equations are used to solve problems involving motion, forces, and energy. For example, in kinematics, systems of equations can be used to determine the position and velocity of an object at a given time. In mechanics, systems of equations are used to analyze the forces acting on an object and determine its acceleration. Consider a scenario involving projectile motion, where the horizontal and vertical components of motion can be described by a system of equations. Solving this system allows physicists to predict the trajectory of the projectile, a crucial application in fields ranging from sports to military science.
Computer Science
In computer science, systems of equations are used in various algorithms and applications, such as linear programming, optimization, and data analysis. For example, linear programming involves optimizing a linear objective function subject to a set of linear constraints, which can be represented as a system of equations and inequalities. This technique is widely used in operations research, logistics, and resource management. Furthermore, systems of equations are used in machine learning algorithms for tasks such as regression and classification. The ability to solve systems of equations is therefore a valuable asset for computer scientists and data analysts.
Conclusion
In summary, the solution to the given system of equations:
y = 1.5x - 4
y = -x
is (1.6, -1.6), corresponding to option D. We arrived at this solution by employing the substitution method, verifying our results, and understanding the graphical interpretation of the equations. Furthermore, we explored the diverse real-world applications of solving systems of equations, highlighting its importance across various disciplines. Mastering the techniques for solving systems of equations not only enhances mathematical proficiency but also equips individuals with valuable problem-solving skills applicable in numerous contexts. Whether it's in economics, engineering, physics, or computer science, the ability to solve systems of equations is a fundamental skill that empowers informed decision-making and effective problem-solving.
