Solving For A+b+c Given A System Of Equations

In the realm of mathematics, particularly in algebra, we often encounter systems of equations that challenge us to find the values of unknown variables. These systems can appear in various forms, sometimes straightforward and sometimes requiring a bit of algebraic manipulation to unravel. This article delves into a classic problem involving a system of three equations with three unknowns: $a$, $b$, and $c$. The given equations are $a + b = 3$, $b + c = 4$, and $c + a = 5$. Our mission is to determine the value of $a + b + c$. This problem not only tests our understanding of basic algebraic principles but also highlights the elegance and efficiency of certain problem-solving techniques. The solution involves a clever approach of adding the equations together and then using substitution to arrive at the final answer. Let's embark on this mathematical journey and explore the steps to solve this intriguing problem.

Setting up the Equations

To begin our exploration, let's first lay out the given equations clearly. We have three equations:

1. $a + b = 3$
2. $b + c = 4$
3. $c + a = 5

These equations form a system of linear equations, a fundamental concept in algebra. Each equation represents a relationship between the variables $a$, $b$, and $c$. Our goal is to find the value of $a + b + c$, which might not be immediately obvious from the individual equations. A common strategy for solving such systems is to manipulate the equations in a way that allows us to isolate or combine variables. In this case, we'll employ a technique that involves adding the equations together. This is a powerful method because it can simplify the system and reveal hidden relationships between the variables. By carefully adding the equations, we aim to create a new equation that directly involves the expression we want to find, which is $a + b + c$. This initial setup is crucial for guiding our problem-solving process and ensuring we proceed in a logical and efficient manner. The beauty of algebra lies in its ability to transform complex problems into manageable steps, and this problem is a perfect example of that.

Adding the Equations

The next crucial step in solving this system of equations is to add all three equations together. This might seem like a simple operation, but it's a pivotal move that unlocks the solution. When we add the left-hand sides of the equations, we get:

$(a + b) + (b + c) + (c + a)$

Similarly, adding the right-hand sides gives us:

$3 + 4 + 5$

Now, let's simplify both sides. On the left-hand side, we can rearrange and combine like terms:

$a + a + b + b + c + c = 2a + 2b + 2c$

This can be further simplified by factoring out the common factor of 2:

$2(a + b + c)$

On the right-hand side, the sum is straightforward:

$3 + 4 + 5 = 12$

So, by adding the equations, we've arrived at a new equation:

$2(a + b + c) = 12$

This equation is a significant milestone in our solution. It directly relates the expression we're interested in, $a + b + c$, to a constant value. The next step will be to isolate $a + b + c$ and find its value. This process of adding equations together is a common technique in algebra, often used to eliminate variables or create simpler equations. It's a testament to the power of algebraic manipulation in solving complex problems.

Solving for a + b + c

Having successfully added the equations, we've arrived at the equation:

$2(a + b + c) = 12$

To find the value of $a + b + c$, we need to isolate this expression. This is a simple algebraic step that involves dividing both sides of the equation by 2. When we divide both sides by 2, we get:

$(2(a + b + c)) / 2 = 12 / 2$

This simplifies to:

$a + b + c = 6$

And there we have it! The value of $a + b + c$ is 6. This result is a direct consequence of the initial equations and the algebraic manipulations we performed. It demonstrates the power of systematic problem-solving in mathematics. By adding the equations and then isolating the desired expression, we were able to find the solution efficiently. This problem highlights the beauty of algebra in its ability to transform seemingly complex relationships into simple, understandable results. The value of $a + b + c$ being 6 is the final answer to our problem, and it concludes our journey through this system of equations.

In this article, we successfully navigated a system of three equations to find the value of $a + b + c$. The given equations were $a + b = 3$, $b + c = 4$, and $c + a = 5$. By employing a clever strategy of adding all three equations together, we arrived at the equation $2(a + b + c) = 12$. A simple division then revealed that $a + b + c = 6$. This problem serves as a testament to the power of algebraic manipulation and systematic problem-solving. The ability to transform and simplify equations is a fundamental skill in mathematics, and this example showcases its effectiveness. The solution not only provides the answer but also demonstrates the elegance and efficiency of algebraic techniques. Understanding how to solve systems of equations is crucial in various fields, from engineering to economics, making this a valuable skill to master. The final answer, $a + b + c = 6$, concludes our exploration of this problem, leaving us with a sense of accomplishment and a deeper appreciation for the beauty of mathematics.