Solving $x + 3y = 7$ 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, particularly crucial in algebra and various applications of mathematical modeling. A system of equations comprises two or more equations with the same set of variables. The solution to a system of equations represents the values that, when substituted for the variables, satisfy all equations simultaneously. In this comprehensive guide, we will delve into a systematic approach to solving the system of equations: $x + 3y = 7$ and $2x + 4y = 8$. By employing the method of substitution, we will navigate through each step, ensuring a clear understanding of the underlying principles and techniques.

1. Isolating $x$ in the First Equation

To initiate the process of solving this system of equations, our primary objective is to isolate $x$ in the first equation, which is $x + 3y = 7$. Isolating a variable entails manipulating the equation algebraically to express that variable in terms of the other variables and constants. In this instance, our goal is to express $x$ as a function of $y$. To achieve this, we subtract $3y$ from both sides of the equation. This operation maintains the equality of the equation while effectively isolating $x$ on one side.

Let's delve into the algebraic manipulation:

$x + 3y = 7$

Subtracting $3y$ from both sides:

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

This simplifies to:

$x = 7 - 3y$

Now, we have successfully isolated $x$ in the first equation. This expression, $x = 7 - 3y$, is crucial as it allows us to substitute this value of $x$ into the second equation, thereby reducing the system to a single equation with one variable, $y$. This strategic step is the cornerstone of the substitution method, enabling us to solve for the variables systematically.

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

Having successfully isolated $x$ in the first equation ($x = 7 - 3y$), the next pivotal step in solving our system of equations is to substitute this expression for $x$ into the second equation. The second equation in our system is $2x + 4y = 8$. By substituting $x$ with $(7 - 3y)$, we effectively eliminate one variable from the second equation, transforming it into an equation solely in terms of $y$. This strategic move is the heart of the substitution method, simplifying the problem and paving the way for us to solve for $y$.

Let's proceed with the substitution:

Starting with the second equation:

$2x + 4y = 8$

Substituting $x$ with $(7 - 3y)$:

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

Now, we have a new equation with only one variable, $y$. This equation can be solved using standard algebraic techniques, which we will explore in the next section.

3. Solving for $y$

Following the substitution of $x$ in the second equation, we now have the equation $2(7 - 3y) + 4y = 8$. Our immediate goal is to solve this equation for $y$. This involves simplifying the equation by distributing and combining like terms, and then isolating $y$ to determine its value. This is a crucial step in our solution process, as the value of $y$ will then be used to find the value of $x$.

Let's break down the algebraic steps:

Starting with the equation:

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

First, distribute the 2 across the terms inside the parentheses:

$14 - 6y + 4y = 8$

Next, combine the like terms (the terms with $y$):

$14 - 2y = 8$

Now, we want to isolate the term with $y$. Subtract 14 from both sides of the equation:

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

This simplifies to:

$-2y = -6$

Finally, divide both sides by -2 to solve for $y$:

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

This gives us:

$y = 3$

We have now successfully found the value of $y$. With $y = 3$, we can substitute this value back into either of the original equations or the expression for $x$ in terms of $y$ to find the value of $x$. This is the next logical step in completing our solution.

4. Solving for $x$

Now that we have determined the value of $y$ to be 3, the subsequent step in solving our system of equations is to solve for $x$. We can accomplish this by substituting the value of $y$ back into one of the original equations or the expression we derived for $x$ in terms of $y$. Opting for the latter often simplifies the process, as $x$ is already isolated.

Recall the expression for $x$ we found in the first step:

$x = 7 - 3y$

Now, substitute $y = 3$ into this expression:

$x = 7 - 3(3)$

Perform the multiplication:

$x = 7 - 9$

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

$x = -2$

Thus, we have found the value of $x$ to be -2. With both $x$ and $y$ determined, we have arrived at the solution to the system of equations.

5. Verifying the Solution

Before finalizing our solution, it is imperative to verify the solution to ensure accuracy. This involves substituting the values we found for $x$ and $y$ back into the original equations to confirm that they satisfy both equations simultaneously. This step serves as a crucial check, guarding against potential errors made during the algebraic manipulations.

Our solution is $x = -2$ and $y = 3$. Let's substitute these values into the original equations:

First equation: $x + 3y = 7$

Substitute $x = -2$ and $y = 3$:

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

Simplify:

$-2 + 9 = 7$

$7 = 7$ (This equation is satisfied)

Second equation: $2x + 4y = 8$

Substitute $x = -2$ and $y = 3$:

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

Simplify:

$-4 + 12 = 8$

$8 = 8$ (This equation is satisfied)

Since the values $x = -2$ and $y = 3$ satisfy both equations, we can confidently conclude that this is the correct solution to the system of equations.

Conclusion

In this detailed guide, we have systematically solved the system of equations $x + 3y = 7$ and $2x + 4y = 8$ using the method of substitution. We began by isolating $x$ in the first equation, expressing it in terms of $y$. Next, we substituted this expression into the second equation, resulting in an equation with only one variable, $y$. We then solved for $y$, obtaining $y = 3$. Subsequently, we substituted the value of $y$ back into the expression for $x$, finding $x = -2$. Finally, we verified our solution by substituting both $x$ and $y$ values into the original equations, confirming that they satisfy both. This step-by-step approach not only provides the solution but also reinforces the understanding of the substitution method, a valuable tool in solving systems of equations. The solution to the system of equations is $x = -2$ and $y = 3$, a testament to the power and precision of algebraic techniques. Understanding and mastering such techniques is paramount for success in mathematics and its myriad applications.