Solving Systems Of Equations The Value Of X Is 1
Introduction
In the realm of mathematics, solving systems of equations is a fundamental skill. This article delves into solving a system of two linear equations to find the value of x. We are presented with the following system:
3x + y = 9
y = -4x + 10
Our goal is to demonstrate that the value of x in this system is indeed 1. We will achieve this by employing the method of substitution, a powerful technique for solving systems of equations. Substitution involves expressing one variable in terms of the other and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, which can then be easily solved. We will meticulously walk through each step of the process, ensuring clarity and understanding. This article serves not only as a solution to the given problem but also as a guide to understanding and applying the substitution method for solving systems of linear equations. Understanding systems of equations is crucial for various applications in mathematics, science, engineering, and economics. They allow us to model real-world situations involving multiple variables and constraints. Mastering the techniques for solving these systems is therefore an invaluable asset. We will also discuss the importance of verifying the solution by substituting the values of both x and y back into the original equations. This step ensures that the solution obtained is indeed correct and satisfies both equations in the system. By the end of this article, you will have a solid understanding of how to solve systems of equations using substitution and be able to apply this knowledge to solve similar problems. Furthermore, you will appreciate the importance of accuracy and verification in mathematical problem-solving. So, let's embark on this mathematical journey and unravel the solution to this system of equations.
1. Substitute the value of y in the first equation
The substitution method is a cornerstone technique in algebra, allowing us to solve systems of equations by expressing one variable in terms of another. In our system, we have two equations:
3x + y = 9
y = -4x + 10
The second equation conveniently expresses y in terms of x. This is a perfect setup for substitution. The core idea is to replace the y in the first equation with its equivalent expression from the second equation. This transforms the first equation into an equation with only one variable, x, which we can then solve directly. The beauty of this method lies in its ability to simplify complex systems into manageable equations. By substituting, we eliminate one variable, effectively reducing the problem's complexity. This step is crucial because it allows us to isolate and determine the value of one variable at a time. Once we find the value of x, we can easily substitute it back into either of the original equations to find the value of y. The process of substitution is not merely a mechanical step; it's a logical deduction that stems from the fundamental principle that if two expressions are equal, one can replace the other without changing the equation's validity. In this specific case, since y is equal to -4x + 10, we can confidently substitute -4x + 10 for y in the first equation. This substitution is the bridge that connects the two equations, allowing us to weave them together and extract the solution. Let's perform the substitution now. Replacing y in the first equation with -4x + 10 gives us:
3x + (-4x + 10) = 9
This equation is now solely in terms of x, setting the stage for the next step where we will combine like terms and simplify the equation further. The substitution has successfully transformed our system into a more manageable form, paving the way for us to determine the value of x.
2. Combine like terms
After performing the substitution, we have the equation:
3x + (-4x + 10) = 9
The next crucial step is to combine like terms. This involves simplifying the equation by grouping together terms that have the same variable and constant terms. In this case, we have two terms with x, namely 3x and -4x, and a constant term 10. Combining like terms is a fundamental algebraic operation that allows us to reduce the complexity of an equation and make it easier to solve. It's based on the distributive property and the idea that we can add or subtract coefficients of terms with the same variable. To combine the x terms, we simply add their coefficients: 3 + (-4) = -1. This means that 3x + (-4x) simplifies to -1x, which is more commonly written as -x. The equation now becomes:
-x + 10 = 9
This equation is significantly simpler than the previous one. We have successfully reduced the number of terms and made the equation more transparent. The goal of combining like terms is not just to simplify the equation but also to isolate the variable we are trying to solve for. By grouping the x terms together, we are one step closer to isolating x on one side of the equation. This process of simplification is essential for solving any algebraic equation. It allows us to manipulate the equation while maintaining its equality, ultimately leading us to the solution. The combined like terms now present a clear picture of the relationship between x and the constants in the equation. We can now proceed to the next step, where we will use the subtraction property of equality to further isolate x. Combining like terms is a fundamental skill in algebra, and mastering it is crucial for success in solving equations and other mathematical problems. It's a technique that is used repeatedly in various mathematical contexts, making it an indispensable tool in any mathematician's arsenal. With the equation simplified, we are now well-positioned to isolate x and find its value.
3. Apply the subtraction property of equality
Having simplified the equation to:
-x + 10 = 9
We now employ the subtraction property of equality. This fundamental property states that we can subtract the same value from both sides of an equation without changing its equality. In other words, if a = b, then a - c = b - c. This property is a cornerstone of algebraic manipulation, allowing us to isolate variables and solve equations. Our goal is to isolate -x on one side of the equation. To do this, we need to eliminate the 10 that is being added to -x. We can achieve this by subtracting 10 from both sides of the equation. This is a direct application of the subtraction property of equality. Subtracting 10 from both sides gives us:
-x + 10 - 10 = 9 - 10
Simplifying this, we get:
-x = -1
We are now very close to finding the value of x. We have successfully isolated -x, but we need to find the value of x itself. To do this, we can multiply both sides of the equation by -1. This is equivalent to changing the sign of both sides. Multiplying both sides by -1 gives us:
(-1) * (-x) = (-1) * (-1)
Simplifying this, we finally arrive at:
x = 1
Thus, we have successfully demonstrated that the value of x in this system of equations is indeed 1. The subtraction property of equality played a crucial role in this process, allowing us to isolate -x and ultimately solve for x. This property, along with other properties of equality, is fundamental to solving algebraic equations. Understanding and applying these properties correctly is essential for success in algebra and beyond. We have now found the value of x, but to complete the solution, we should also find the value of y and verify that our solution satisfies both equations in the system. This will ensure that our solution is correct and complete.
Conclusion
In conclusion, we have successfully demonstrated that the value of x in the given system of equations is 1. We achieved this by employing the method of substitution, a powerful technique for solving systems of equations. We first substituted the expression for y from the second equation into the first equation. This resulted in an equation with only one variable, x. We then combined like terms to simplify the equation and applied the subtraction property of equality to isolate -x. Finally, we multiplied both sides of the equation by -1 to solve for x, obtaining the result x = 1. This process highlights the importance of understanding and applying fundamental algebraic principles. The method of substitution is a versatile technique that can be used to solve a wide variety of systems of equations. It is a testament to the power of algebraic manipulation and the ability to transform complex problems into simpler ones. Furthermore, we emphasized the importance of verifying the solution. While we have shown that x = 1, it is crucial to also find the corresponding value of y and substitute both values back into the original equations to ensure they are satisfied. This step is essential for ensuring the accuracy of the solution. Solving systems of equations is a fundamental skill in mathematics with applications in various fields, including science, engineering, and economics. Mastering these techniques is crucial for anyone pursuing studies or careers in these areas. This article has provided a detailed walkthrough of the solution process, highlighting the key steps and principles involved. By understanding these principles and practicing the techniques, you can confidently tackle similar problems and expand your mathematical problem-solving abilities. The journey of solving this system of equations has not only provided us with the answer but also reinforced the importance of logical reasoning, algebraic manipulation, and the verification of solutions. These are invaluable skills that will serve you well in your mathematical endeavors.
