Solving Simultaneous Equations Step By Step Guide
Introduction
In the realm of mathematics, solving simultaneous equations is a fundamental skill. It is crucial for various applications in science, engineering, economics, and computer science. Simultaneous equations, also known as systems of equations, involve two or more equations with two or more variables. The solution to a system of simultaneous equations is a set of values for the variables that satisfy all equations simultaneously. This article delves into the methods for solving simultaneous equations, focusing on a specific example to illustrate the process.
The ability to solve simultaneous equations is not just an academic exercise; it's a practical tool that enables us to model and analyze real-world scenarios. From determining the equilibrium point in supply and demand curves to calculating the forces acting on a structure, simultaneous equations provide a framework for understanding and solving complex problems. Mastering the techniques for solving these equations is therefore an essential step in developing mathematical proficiency and applying it to diverse fields.
The specific example we will address in this article involves two linear equations with two variables. While this is a relatively simple case, the underlying principles and methods can be extended to systems with more equations and variables. We will explore two common methods for solving such systems: the substitution method and the elimination method. By carefully examining these methods and applying them to our example, we will gain a deeper understanding of the process of solving simultaneous equations.
Problem Statement
Let's consider the following system of simultaneous equations:
4y - 9x = 29
6y - 7x = 24
Our goal is to find the values of x and y that satisfy both equations simultaneously. This means we are looking for a pair of numbers (x, y) that, when substituted into both equations, make both equations true. There are several methods to achieve this, and we will explore two of the most common ones: the substitution method and the elimination method.
The challenge in solving simultaneous equations lies in the fact that each equation individually has infinitely many solutions. For example, in the first equation, 4y - 9x = 29, we could choose any value for x and then solve for y, or vice versa. However, we are not looking for just any solution; we are looking for the solution that works for both equations. This constraint significantly narrows down the possibilities and allows us to pinpoint the unique values of x and y that satisfy the entire system.
Before we delve into the methods, it's worth noting that the graphical interpretation of this problem involves finding the point of intersection of two lines. Each equation represents a line in the xy-plane, and the solution to the system corresponds to the coordinates of the point where the lines intersect. If the lines are parallel, there is no solution; if the lines are the same, there are infinitely many solutions. However, for this particular problem, we expect to find a unique solution, representing a single point of intersection.
Method 1: Substitution
The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with a single variable, which can then be solved. Once we have the value of one variable, we can substitute it back into either of the original equations to find the value of the other variable.
Let's apply the substitution method to our system:
4y - 9x = 29
6y - 7x = 24
First, we choose one of the equations and solve for one of the variables. Let's solve the first equation for y:
4y = 9x + 29
y = (9x + 29) / 4
Now, we substitute this expression for y into the second equation:
6((9x + 29) / 4) - 7x = 24
This equation now involves only the variable x. We can simplify and solve for x:
(54x + 174) / 4 - 7x = 24
54x + 174 - 28x = 96
26x = -78
x = -3
Now that we have the value of x, we can substitute it back into the expression we found for y:
y = (9(-3) + 29) / 4
y = (-27 + 29) / 4
y = 2 / 4
y = 0.5
Therefore, the solution to the system of equations using the substitution method is x = -3 and y = 0.5. We can verify this solution by substituting these values back into the original equations to ensure they are satisfied. This method is particularly useful when one of the equations can be easily solved for one variable in terms of the other.
Method 2: Elimination
The elimination method involves manipulating the equations so that the coefficients of one of the variables are opposites. This allows us to eliminate that variable by adding the equations together. The resulting equation will have only one variable, which can then be solved. Once we have the value of one variable, we can substitute it back into either of the original equations to find the value of the other variable.
Let's apply the elimination method to our system:
4y - 9x = 29
6y - 7x = 24
To eliminate y, we need to make the coefficients of y opposites. We can do this by multiplying the first equation by 3 and the second equation by -2:
3(4y - 9x) = 3(29) => 12y - 27x = 87
-2(6y - 7x) = -2(24) => -12y + 14x = -48
Now, we add the two equations together:
(12y - 27x) + (-12y + 14x) = 87 + (-48)
-13x = 39
x = -3
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 first equation:
4y - 9(-3) = 29
4y + 27 = 29
4y = 2
y = 0.5
Therefore, the solution to the system of equations using the elimination method is x = -3 and y = 0.5. This matches the solution we found using the substitution method, which is a good way to check our work. The elimination method is often preferred when the coefficients of one of the variables are easily made opposites by multiplication.
Verification of the Solution
To ensure the accuracy of our solution, it's crucial to verify the solution by substituting the values of x and y back into the original equations. This step helps us catch any potential errors made during the solving process. If the values satisfy both equations, we can confidently conclude that we have found the correct solution.
Our solution is x = -3 and y = 0.5. Let's substitute these values into the first equation:
4y - 9x = 29
4(0.5) - 9(-3) = 29
2 + 27 = 29
29 = 29
The first equation is satisfied. Now, let's substitute the values into the second equation:
6y - 7x = 24
6(0.5) - 7(-3) = 24
3 + 21 = 24
24 = 24
The second equation is also satisfied. Since the values x = -3 and y = 0.5 satisfy both equations, we can confidently conclude that this is the correct solution to the system of simultaneous equations.
Verifying the solution is an important practice in mathematics, especially when dealing with equations and systems of equations. It provides a sense of certainty and helps in building confidence in the problem-solving process. Moreover, it reinforces the understanding of what it means for a solution to satisfy an equation or a system of equations.
Conclusion
In this article, we have explored the methods for solving simultaneous equations, focusing on the specific example:
4y - 9x = 29
6y - 7x = 24
We successfully found the solution x = -3 and y = 0.5 using both the substitution and elimination methods. This demonstrates the versatility of these techniques and the importance of choosing the most efficient method for a given problem.
Solving simultaneous equations is a fundamental skill in mathematics with wide-ranging applications. The ability to manipulate equations, isolate variables, and verify solutions is crucial for success in various fields. Whether it's modeling physical systems, analyzing economic trends, or solving engineering problems, simultaneous equations provide a powerful tool for understanding and making predictions about the world around us.
Moreover, the process of solving simultaneous equations fosters critical thinking and problem-solving skills. It requires a systematic approach, attention to detail, and the ability to apply mathematical concepts in a logical manner. The skills developed through solving these equations are transferable to other areas of study and professional endeavors.
In conclusion, mastering the techniques for solving simultaneous equations is an investment in one's mathematical abilities and a valuable asset for tackling real-world challenges. By understanding the underlying principles and practicing the methods, individuals can confidently approach and solve complex problems involving multiple variables and relationships.
