Solving Systems Of Equations A Step-by-Step Guide To X + 12y = 68 And X = 8y - 12

by Jeany 82 views
Iklan Headers

In the realm of mathematics, solving systems of equations is a fundamental skill. This article will provide a comprehensive, step-by-step guide on how to solve the system of equations:

\begin{cases}
x + 12y = 68 \\
x = 8y - 12
\end{cases}

We'll delve into the methods and techniques required to find the values of x and y that satisfy both equations simultaneously. This exploration is crucial not only for academic pursuits but also for real-world applications where interconnected variables need to be determined. Let's embark on this mathematical journey together and unravel the solution to this system.

Understanding Systems of Equations

Before diving into the solution, it's crucial to grasp the underlying concept of a system of equations. Systems of equations represent a set of two or more equations containing multiple variables. The solution to a system is the set of values for the variables that make all equations in the system true simultaneously. In simpler terms, we're looking for the point where the lines represented by these equations intersect on a graph. The system we're tackling today involves two linear equations, each with two variables (x and y). This type of system can have one solution, infinitely many solutions (if the equations represent the same line), or no solution (if the lines are parallel). The beauty of algebra lies in its ability to systematically determine which of these scenarios we're facing and, if a solution exists, to pinpoint its exact values. Mastering the art of solving systems of equations opens doors to modeling and solving real-world problems in diverse fields like physics, engineering, economics, and computer science. So, understanding the fundamentals is the first step toward wielding this powerful mathematical tool.

Methods for Solving Systems of Equations

Several methods exist for solving systems of equations, each with its strengths and weaknesses. Understanding these methods allows us to choose the most efficient approach for a given problem. The two primary methods are:

  1. Substitution Method: This involves solving one equation for one variable and substituting that expression into the other equation. This eliminates one variable, allowing us to solve for the remaining one. Once we have the value of one variable, we can substitute it back into either original equation to find the value of the other variable. The substitution method is particularly useful when one equation is already solved for one variable, or when it's easy to isolate one variable.

  2. Elimination Method (also known as the Addition Method): This method involves manipulating the equations so that the coefficients of one variable are opposites. By adding the equations together, this variable is eliminated, leaving us with a single equation in one variable. We can then solve for that variable and substitute back into one of the original equations to find the other variable. The elimination method is often preferred when the coefficients of one variable are easily made opposites through multiplication.

Beyond these two core methods, graphical methods can also provide insights into the solutions of a system. By graphing the equations, the point(s) of intersection represent the solution(s) to the system. However, graphical methods can be less precise for finding exact solutions, especially when the solutions are not integers. In more advanced contexts, matrix methods are employed for solving larger systems of equations, offering a compact and efficient approach. Choosing the appropriate method often depends on the specific structure of the equations and the desired level of precision.

Solving the System Using Substitution

Let's apply the substitution method to solve the system:

\begin{cases}
x + 12y = 68 \\
x = 8y - 12
\end{cases}

The second equation, x = 8y - 12, is already solved for x. This makes the substitution method a natural choice. We'll substitute the expression for x from the second equation into the first equation.

  1. Substitute x in the first equation: Replace x in the equation x + 12y = 68 with the expression 8y - 12. This gives us: (8y - 12) + 12y = 68.

  2. Simplify and solve for y: Combine like terms: 8y + 12y - 12 = 68, which simplifies to 20y - 12 = 68. Add 12 to both sides: 20y = 80. Divide both sides by 20: y = 4.

  3. Substitute y back into an equation to find x: We can use either of the original equations. The second equation, x = 8y - 12, is slightly simpler. Substitute y = 4: x = 8(4) - 12 = 32 - 12 = 20.

Therefore, the solution to the system is x = 20 and y = 4. This means the point (20, 4) is the intersection of the two lines represented by the equations. The substitution method elegantly transforms a system of two equations into a single equation, allowing us to isolate and solve for one variable at a time. This step-by-step process underscores the power of algebraic manipulation in finding solutions to seemingly complex problems.

Solving the System Using Elimination

Now, let's solve the same system using the elimination method to demonstrate an alternative approach and reinforce our understanding. The system is:

\begin{cases}
x + 12y = 68 \\
x = 8y - 12
\end{cases}

To use the elimination method effectively, we need to align the variables and manipulate the equations so that the coefficients of either x or y are opposites.

  1. Rewrite the second equation: To align the variables, rewrite the second equation as x - 8y = -12. Now our system looks like this:
\begin{cases}
x + 12y = 68 \\
x - 8y = -12
\end{cases}

  1. Eliminate x: Notice that the coefficients of x are already the same (both are 1). To eliminate x, we can multiply the second equation by -1 and then add the equations together. Multiplying the second equation by -1 gives us -x + 8y = 12.

  2. Add the equations: Add the modified second equation to the first equation:

(x + 12y) + (-x + 8y) = 68 + 12

This simplifies to 20y = 80.

  1. Solve for y: Divide both sides by 20: y = 4.

  2. Substitute y back into an equation to find x: We can use either of the original equations. Let's use the first equation, x + 12y = 68. Substitute y = 4: x + 12(4) = 68, which simplifies to x + 48 = 68. Subtract 48 from both sides: x = 20.

As we found using the substitution method, the solution to the system is x = 20 and y = 4. The elimination method cleverly manipulates the equations to cancel out one variable, streamlining the process of finding the solution. This alternative approach underscores the flexibility and power of algebraic techniques in solving systems of equations.

Verifying the Solution

After obtaining a solution, it's crucial to verify the solution to ensure accuracy. This step confirms that the values we found for x and y satisfy both equations in the system. To verify our solution (x = 20, y = 4), we'll substitute these values back into the original equations:

  1. First equation: x + 12y = 68. Substitute x = 20 and y = 4: 20 + 12(4) = 20 + 48 = 68. This equation holds true.

  2. Second equation: x = 8y - 12. Substitute x = 20 and y = 4: 20 = 8(4) - 12 = 32 - 12 = 20. This equation also holds true.

Since the values x = 20 and y = 4 satisfy both equations, we can confidently conclude that this is the correct solution to the system. Verification is a vital step in the problem-solving process, particularly in mathematics. It minimizes the risk of errors and ensures that the obtained solution is indeed valid within the context of the given equations. By systematically checking our work, we strengthen our understanding and build confidence in our problem-solving abilities.

Applications of Systems of Equations

Systems of equations are not just abstract mathematical concepts; they have a wide range of practical applications in various fields. Understanding these applications highlights the real-world significance of mastering the techniques for solving systems of equations.

  1. Real-World Problem Solving: Systems of equations can model and solve problems involving multiple variables and constraints. For instance, in economics, they can be used to determine equilibrium prices and quantities in markets. In physics, they can be used to analyze the motion of objects under multiple forces. In engineering, they can be used to design structures and circuits. Consider a scenario where you need to determine the amount of two different investments to make to reach a specific financial goal, given different interest rates and timeframes. This can be elegantly solved using a system of equations.

  2. Science and Engineering: In fields like physics and engineering, systems of equations are fundamental for modeling physical systems. For example, circuit analysis involves solving systems of equations to determine currents and voltages in different parts of a circuit. Chemical reactions can also be modeled using systems of equations to determine the amounts of reactants and products at equilibrium. Furthermore, in structural engineering, systems of equations are crucial for calculating forces and stresses within structures to ensure stability and safety.

  3. Computer Graphics and Game Development: Systems of equations play a crucial role in computer graphics and game development. They are used to perform transformations such as rotations, scaling, and translations of objects in 3D space. Solving systems of equations is also essential for collision detection and response in games, ensuring that objects interact realistically. The ability to manipulate and solve these equations efficiently is vital for creating visually appealing and interactive experiences.

  4. Data Analysis and Statistics: In data analysis and statistics, systems of equations are used in regression analysis to find the best-fit line or curve for a set of data points. This allows us to model relationships between variables and make predictions. They are also used in solving systems of linear equations that arise in various statistical models, such as those used in econometrics and social sciences. The application of these techniques allows for a deeper understanding of complex datasets and the extraction of meaningful insights.

In conclusion, the ability to solve systems of equations is a valuable skill that extends far beyond the classroom. From optimizing business decisions to modeling complex physical phenomena, these mathematical tools provide a powerful framework for understanding and solving real-world problems.

Conclusion

In this comprehensive guide, we've explored the process of solving the system of equations:

\begin{cases}
x + 12y = 68 \\
x = 8y - 12
\end{cases}

We've demonstrated two primary methods – substitution and elimination – and showed how both lead to the same solution: x = 20 and y = 4. We also emphasized the importance of verifying the solution to ensure accuracy. Furthermore, we've highlighted the diverse applications of systems of equations in various fields, underscoring their practical significance.

Mastering the art of solving systems of equations is a cornerstone of mathematical proficiency. It not only equips you with a valuable problem-solving skill but also opens doors to understanding and modeling complex phenomena in the real world. As you continue your mathematical journey, remember that practice and perseverance are key to mastering these techniques. The ability to approach problems systematically, choose the appropriate method, and verify your solutions will serve you well in your academic and professional pursuits. So, embrace the challenge, delve into the world of equations, and unlock the power of mathematical thinking.