Solving For X In A System Of Equations A Step-by-Step Guide
Introduction
In mathematics, solving systems of equations is a fundamental skill. Systems of equations are sets of two or more equations containing the same variables. The solution to a system of equations is the set of values for the variables that make all the equations true simultaneously. There are several methods to solve systems of equations, including substitution, elimination, and graphing. In this article, we'll delve into solving a specific system of linear equations using the substitution method, highlighting each step to find the value of x. Our system of equations is:
3x + y = 9
y = -4x + 10
We aim to find the value of x that satisfies both equations. The substitution method is particularly useful when one of the equations is already solved for one variable in terms of the other, as is the case here with the second equation expressing y in terms of x. This method involves substituting the expression for one variable from one equation into the other equation, effectively reducing the system to a single equation with one variable. By solving this equation, we find the value of one variable, which can then be substituted back into either of the original equations to find the value of the other variable. This process ensures that we find the values of x and y that make both equations true, providing the solution to the system. Mastering the substitution method is crucial for solving more complex systems of equations and for understanding various mathematical concepts that rely on algebraic manipulation. The efficiency and versatility of the substitution method make it a valuable tool in algebra and beyond. Understanding the underlying principles of this method allows students and professionals alike to tackle a wide range of problems in mathematics, science, and engineering.
Step 1: Substitute the value of y in the first equation
The first step in solving this system using the substitution method involves taking the expression for y from the second equation and substituting it into the first equation. This is a critical step because it reduces the system of two equations with two variables into a single equation with one variable, making it solvable. The second equation gives us y = -4x + 10. Now, we substitute this expression for y into the first equation, which is 3x + y = 9. By replacing y with -4x + 10, the first equation becomes:
3x + (-4x + 10) = 9
This substitution is a fundamental algebraic technique that allows us to eliminate one variable and focus on solving for the remaining one. It’s like taking a piece from one puzzle and fitting it into another, simplifying the overall problem. The new equation, 3x + (-4x + 10) = 9, now contains only the variable x, which means we can use algebraic manipulations to isolate x and find its value. The substitution process is not just about replacing one symbol with another; it's about transforming the equation in a way that makes it easier to solve. This step is crucial because it sets the stage for the rest of the solution. Without this initial substitution, we would still have a system of two equations with two unknowns, which is more challenging to solve directly. The ability to perform this substitution accurately and efficiently is a key skill in algebra and is essential for solving more complex mathematical problems. Understanding the rationale behind this step – reducing the number of variables – is just as important as the mechanics of the substitution itself.
Step 2: Combine Like Terms
After substituting the value of y into the first equation, we have 3x + (-4x + 10) = 9. The next step is to combine like terms to simplify the equation. Like terms are terms that contain the same variable raised to the same power. In this equation, 3x and -4x are like terms because they both contain the variable x raised to the power of 1. To combine these terms, we simply add their coefficients. The coefficient of 3x is 3, and the coefficient of -4x is -4. Adding these coefficients gives us 3 + (-4) = -1. Therefore, combining 3x and -4x results in -1x, which is commonly written as -x. The equation now becomes:
-x + 10 = 9
This simplification is a crucial step because it makes the equation easier to solve. Combining like terms reduces the number of terms in the equation, making it more manageable. The process of combining like terms is a fundamental algebraic skill that is used extensively in solving equations and simplifying expressions. It involves identifying terms that share the same variable and exponent, and then adding or subtracting their coefficients. The constant term, 10, remains unchanged at this step because there are no other constant terms to combine it with. The simplified equation, -x + 10 = 9, is now in a form that is much easier to work with. We have reduced the equation to a simple linear equation with one variable, which can be solved using basic algebraic operations. This step highlights the importance of simplification in mathematics. By combining like terms, we reduce the complexity of the equation and pave the way for the final steps in solving for x. The ability to accurately combine like terms is a cornerstone of algebraic manipulation and is essential for success in more advanced mathematical topics.
Step 3: Apply the Subtraction Property of Equality
After combining like terms, our equation is -x + 10 = 9. The next step involves applying the subtraction property of equality. This property states that you can subtract the same value from both sides of an equation without changing the equality. In this case, we want to isolate the term with x, which is -x. To do this, we need to eliminate the constant term, 10, from the left side of the equation. We can achieve this by subtracting 10 from both sides of the equation:
-x + 10 - 10 = 9 - 10
This simplifies to:
-x = -1
The subtraction property of equality is a fundamental principle in algebra. It ensures that the equation remains balanced while we manipulate it to solve for the variable. By subtracting 10 from both sides, we maintain the equality while moving closer to isolating x. This step is crucial because it isolates the term containing x, making it easier to solve for x itself. The equation -x = -1 indicates that the negative of x is equal to -1. To find the value of x, we need to eliminate the negative sign. We can do this by multiplying both sides of the equation by -1, which is equivalent to changing the sign of both sides. Applying the subtraction property of equality is a common technique in solving algebraic equations. It’s a powerful tool that allows us to manipulate equations while preserving their balance and accuracy. Understanding and applying this property correctly is essential for solving a wide range of mathematical problems. The next step, as we’ll see, involves dealing with the negative sign in front of x to find the final value of x.
Step 4: Solve for x
Following the subtraction property of equality, we arrived at the equation -x = -1. To find the value of x, we need to eliminate the negative sign. This can be achieved by multiplying both sides of the equation by -1. This is equivalent to changing the sign of both sides:
(-1) * (-x) = (-1) * (-1)
This simplifies to:
x = 1
Therefore, the value of x that satisfies the system of equations is 1. This final step is crucial because it provides the solution to the problem. After performing the substitution, combining like terms, and applying the subtraction property of equality, we have successfully isolated x and determined its value. The process of multiplying both sides of the equation by -1 is a simple yet important algebraic manipulation. It allows us to transform -x into x, giving us the direct value of the variable we are solving for. The solution x = 1 means that when x is equal to 1, both equations in the system are true. This value of x is part of the solution to the system of equations. To find the complete solution, we would also need to find the value of y by substituting x = 1 into either of the original equations. However, our focus here was on finding the value of x, which we have successfully done. This step demonstrates the importance of algebraic manipulation in solving equations. By applying the correct operations to both sides of the equation, we can isolate the variable and find its value. The solution x = 1 is the answer we were looking for, and it represents a key component of the solution to the system of equations.
Conclusion
In conclusion, by using the substitution method and applying algebraic principles such as combining like terms and the subtraction property of equality, we have successfully determined that the value of x in the given system of equations is 1. The steps involved – substituting the expression for y, combining like terms, applying the subtraction property, and solving for x – demonstrate a systematic approach to solving systems of equations. This method is a powerful tool in algebra and is applicable to a wide range of problems. Understanding and mastering these techniques is essential for anyone studying mathematics or related fields. The ability to solve systems of equations is not only a fundamental skill in algebra but also has applications in various real-world scenarios. From modeling physical systems to solving economic problems, the ability to find solutions to systems of equations is invaluable. The process we followed highlights the importance of careful algebraic manipulation and the logical progression of steps required to arrive at the correct solution. The value of x = 1 is a crucial piece of the puzzle, and it represents the x-coordinate of the point where the two lines represented by the equations intersect. The substitution method, as demonstrated in this article, provides a clear and effective way to find these solutions. By breaking down the problem into manageable steps, we can solve even complex systems of equations with confidence. The key is to understand the underlying principles and apply them systematically. The solution x = 1 is not just a number; it is a piece of information that has meaning in the context of the system of equations. It is a testament to the power of algebra and the ability to solve problems using logical reasoning and mathematical techniques.
