Solving Systems Of Equations $x + 3y = 7$ And $2x + 4y = 8$ A Step-by-Step Guide

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In the realm of mathematics, solving systems of equations is a fundamental skill. This article will delve into a step-by-step method to solve the system of equations: $x + 3y = 7$ and $2x + 4y = 8$. Understanding how to solve these systems is crucial for various applications in science, engineering, and economics. We will break down each step, ensuring clarity and comprehension.

1. Isolating $x$ in the First Equation

To begin solving the system of equations, we first need to isolate one variable in one of the equations. Isolating x in the first equation, $x + 3y = 7$, is a straightforward process. Our goal is to express $x$ in terms of $y$. To achieve this, we subtract $3y$ from both sides of the equation. This operation maintains the balance of the equation while moving the $3y$ term to the right side.

Subtracting $3y$ from both sides, we get:

$x + 3y - 3y = 7 - 3y$

Simplifying the equation, we find:

$x = 7 - 3y$

This equation now expresses $x$ explicitly in terms of $y$. This isolated expression for $x$ is a crucial stepping stone. This expression will be substituted into the second equation, allowing us to solve for $y$. The ability to isolate variables is a cornerstone of algebraic manipulation, and mastering this skill is essential for solving more complex problems. Furthermore, this method, known as the substitution method, is widely applicable to various systems of equations, making it a valuable tool in any mathematician's arsenal. The clear and concise isolation of $x$ sets the stage for the next step, where we leverage this expression to unravel the value of $y$ and subsequently find the value of $x$.

2. Substituting the Value of $x$ into the Second Equation

Having successfully isolated x in the first equation ($x = 7 - 3y$), the next critical step is to substitute this expression into the second equation, which is $2x + 4y = 8$. Substitution is a powerful technique in solving systems of equations because it allows us to reduce the system to a single equation with a single variable. This simplification makes the equation solvable.

Replacing $x$ with $(7 - 3y)$ in the second equation, we get:

$2(7 - 3y) + 4y = 8$

This substitution effectively eliminates $x$ from the second equation, leaving us with an equation solely in terms of $y$. The next task is to simplify this equation by distributing the 2 and combining like terms. This process is crucial for isolating $y$ and finding its value. The act of substitution is not just a mechanical step but a strategic move to transform a two-variable problem into a single-variable one. It showcases the interconnectedness of the equations and how manipulating one can provide insights into the other. By carefully substituting the expression for $x$, we pave the way for solving for $y$, bringing us closer to the solution of the entire system of equations. The accuracy of this substitution is paramount, as any error here will propagate through the rest of the solution.

3. Solving for $y$

Following the substitution of $x$ in the second equation, we arrived at $2(7 - 3y) + 4y = 8$. The immediate next step is to distribute the 2 across the parentheses. This involves multiplying both terms inside the parentheses by 2, which is a fundamental application of the distributive property.

Distributing the 2, we get:

$14 - 6y + 4y = 8$

Now, we need to combine the like terms involving $y$. We have $-6y$ and $+4y$, which can be combined to simplify the equation further. Combining like terms is a crucial step in solving equations, as it reduces the complexity and brings us closer to isolating the variable of interest.

Combining the $y$ terms, we have:

$14 - 2y = 8$

Our next goal is to isolate the term with $y$ on one side of the equation. To do this, we subtract 14 from both sides of the equation. This maintains the balance of the equation while moving the constant term to the right side.

Subtracting 14 from both sides gives:

$14 - 2y - 14 = 8 - 14$

Simplifying, we get:

$-2y = -6$

Finally, to solve for $y$, we divide both sides of the equation by -2. This isolates $y$ and gives us its value.

Dividing both sides by -2:

$y = \frac{-6}{-2}$

Simplifying, we find:

$y = 3$

Thus, we have successfully solved for $y$, finding that $y = 3$. This value is a critical piece of the solution to the system of equations. With the value of $y$ in hand, we can now proceed to find the value of $x$. The process of isolating y involved multiple algebraic manipulations, each carefully executed to maintain the equation's integrity and lead us closer to the solution. The result, $y = 3$, is a testament to the power of these techniques and sets the stage for the final step in solving the system.

4. Solving for $x$

Now that we have determined the value of $y$ to be 3, we can substitute this value back into either of the original equations or the isolated equation $x = 7 - 3y$ to solve for $x$. The isolated equation is often the most convenient choice as it directly expresses $x$ in terms of $y$. This step solidifies the solution to the system by providing the corresponding value of $x$ for the found value of $y$.

Substituting $y = 3$ into the equation $x = 7 - 3y$, we get:

$x = 7 - 3(3)$

Now, we perform the multiplication:

$x = 7 - 9$

Finally, we subtract to find the value of $x$:

$x = -2$

Therefore, we have found that $x = -2$. This completes the solution to the system of equations. The value of x, in conjunction with the previously found value of $y$, provides a complete solution that satisfies both equations in the system. The consistency of this solution can be verified by substituting both $x = -2$ and $y = 3$ into both original equations and confirming that they hold true. This final step of solving for $x$ underscores the power of the substitution method and highlights the interconnectedness of variables in a system of equations. The successful determination of $x$ concludes our journey through the solution process, leaving us with a definitive answer to the problem.

5. Verifying the Solution

To ensure the accuracy of our solution, it is crucial to verify the values of $x$ and $y$ in both original equations. This step acts as a safeguard against potential errors made during the solving process. Verification involves substituting the obtained values back into the original equations and confirming that both equations hold true. This not only provides confidence in the solution but also reinforces the understanding of what it means to solve a system of equations.

Our solution is $x = -2$ and $y = 3$. Let's substitute these values into the first equation, $x + 3y = 7$:

$(-2) + 3(3) = 7$

Simplifying, we get:

$-2 + 9 = 7$

$7 = 7$

The first equation holds true. Now, let's substitute the values into the second equation, $2x + 4y = 8$:

$2(-2) + 4(3) = 8$

Simplifying, we get:

$-4 + 12 = 8$

$8 = 8$

The second equation also holds true. Since both equations are satisfied by our solution, we can confidently conclude that $x = -2$ and $y = 3$ is the correct solution to the system of equations. The act of verifying the solution underscores the importance of precision and attention to detail in mathematics. It provides a sense of closure and confirms that the steps taken throughout the solution process were accurate and led to the correct answer. This verification process is not merely a formality but a critical component of problem-solving, reinforcing the validity of the solution and enhancing comprehension of the underlying mathematical principles.

Conclusion

In summary, we have successfully navigated through the process of solving the system of equations $x + 3y = 7$ and $2x + 4y = 8$. By employing the substitution method, we first isolated x in the first equation, then substituted this expression into the second equation to solve for $y$. Once we found the value of $y$, we substituted it back into the isolated equation to determine the value of $x$. Finally, we verified our solution by plugging the values of $x$ and $y$ back into both original equations to ensure their validity. This step-by-step approach highlights the systematic nature of solving systems of equations and provides a clear roadmap for tackling similar problems in the future. The ability to solve systems of equations is a fundamental skill in mathematics with wide-ranging applications. Mastering this skill not only enhances problem-solving abilities but also deepens understanding of algebraic concepts. The detailed walkthrough provided in this article serves as a valuable resource for students and anyone seeking to strengthen their mathematical prowess. The careful execution of each step, from isolating variables to verifying the solution, underscores the importance of precision and attention to detail in mathematics. The successful resolution of this system of equations stands as a testament to the power of algebraic techniques and their ability to unravel complex relationships between variables.