Solving 3x-2y=6 And X-4y=-8 With Elimination Method A Comprehensive Guide
Hey guys! Ever stumbled upon a math problem that looks like a jumbled mess of letters and numbers? Don't worry, we've all been there. Today, we're going to tackle a classic: solving systems of linear equations. Specifically, we'll be diving deep into the elimination method, a powerful technique to find the solutions when you have two or more equations with two or more unknowns. So, grab your pencils, and let's get started!
What are Systems of Equations?
Before we jump into the elimination method, let's quickly recap what systems of equations are. Imagine you have two equations, each representing a line on a graph. The solution to the system is the point where these lines intersect – the (x, y) coordinates that satisfy both equations simultaneously. These problems are common across various fields, from engineering to economics, where multiple factors influence each other. The beauty of the elimination method lies in its systematic approach to unravel these interdependent relationships, offering a clear path to the solution. Whether you're a student grappling with algebra or a professional applying quantitative methods, understanding this technique is a valuable asset.
Systems of equations pop up everywhere in real life. Think about it: maybe you're trying to figure out the cost of two different items given their combined price and a separate relationship between their individual prices. Or perhaps you're modeling the motion of objects in physics, where multiple forces are acting at once. In mathematics, a system of equations is a set of two or more equations that share the same variables. Our goal is to find the values of these variables that make all the equations true at the same time. The elimination method provides a structured approach to solve these systems, making complex problems manageable and understandable. As we proceed, remember that each step we take is designed to simplify the equations, bringing us closer to the solution with clarity and precision.
Linear equations are a particular type of equation where the variables are only raised to the power of 1 (no squares, cubes, etc.). Graphically, they represent straight lines. The problems we are going to discuss are focused on linear equations, where the highest power of the variables is one, which ensures that our equations form straight lines when graphed. This linearity simplifies the solution process, allowing us to use methods like elimination effectively. The core idea is to manipulate these equations in a way that we can eliminate one variable, making it easier to solve for the other. This approach is not only efficient but also provides a clear, step-by-step path to finding the values that satisfy all equations in the system. By understanding linear equations, you gain a fundamental tool for solving a wide range of mathematical and real-world problems. So, let's keep this definition in mind as we tackle our example and unlock the solution.
Meet Our Equations: 3x - 2y = 6 and x - 4y = -8
Okay, let's get down to business. We have the following system of equations:
- 3x - 2y = 6
- x - 4y = -8
Our mission, should we choose to accept it (and we do!), is to find the values of x and y that satisfy both of these equations. In this scenario, we're presented with two linear equations, each containing two variables: x and y. The challenge lies in finding the specific values for these variables that make both equations true simultaneously. This is where the elimination method comes into play, offering a strategic approach to untangle these relationships. The method hinges on manipulating the equations so that either the x or y terms cancel out when the equations are added or subtracted. This simplification allows us to solve for one variable at a time, making the problem more manageable and paving the way for a clear solution. As we delve deeper into the steps, you'll see how each manipulation is a deliberate move toward isolating the variables and revealing their values.
These equations represent two straight lines on a graph. The point where these lines intersect is the solution to our system. Visually, each equation can be thought of as a line stretching across a coordinate plane, and the solution we seek is the precise point where these lines cross paths. This intersection point is special because its x and y coordinates satisfy both equations—it's the single spot where the two lines agree. The elimination method is a technique that helps us find this point algebraically, without needing to graph the lines. It's a powerful tool for solving systems of equations because it breaks down the problem into manageable steps, allowing us to systematically eliminate one variable and solve for the other. By finding this intersection, we uncover the values of x and y that make both equations true, providing a complete solution to our system.
We're going to use the elimination method to solve this. The idea behind the elimination method is to manipulate the equations so that when we add or subtract them, one of the variables disappears. It's like magic, but with algebra! The elimination method is a strategic approach in algebra designed to simplify the process of solving systems of equations. The core principle involves altering the equations in such a way that when they are combined—either by addition or subtraction—one of the variables is effectively removed from the equation. This clever maneuver reduces the complexity of the system, allowing us to focus on solving for the remaining variable. The power of this method lies in its ability to systematically break down a complex problem into simpler, more manageable steps. By eliminating one variable, we pave the way for a straightforward calculation of the other, bringing us closer to the solution with each manipulation.
Step 1: Setting Up for Elimination
Looking at our equations, we can see that if we multiply the second equation by -3, the 'x' terms will be opposites (3x and -3x). This is the key to elimination! The goal of the first step in the elimination method is to prepare our equations so that we can easily eliminate one of the variables. This often involves multiplying one or both equations by a constant, a strategic move that sets the stage for cancellation. By carefully choosing our multipliers, we aim to make the coefficients of either x or y opposites. This ensures that when we add the equations together, one variable will vanish, simplifying the system. The beauty of this step lies in its flexibility; we can tailor our approach to best suit the equations at hand, setting the groundwork for a smooth and efficient solution. Think of it as laying the foundation for a mathematical masterpiece, where each manipulation brings us closer to the final answer.
Let's do it:
-3 * (x - 4y) = -3 * (-8)
This gives us:
-3x + 12y = 24
Now we have our modified second equation. The act of multiplying the entire equation by a constant is a crucial step in the elimination method, ensuring that the equality of the equation remains intact while strategically altering its form. This maneuver is not just about changing numbers; it's about transforming the equation to align with our goal of eliminating a variable. By distributing the constant across all terms, we create an equivalent equation that is perfectly poised to help us cancel out a variable when combined with another equation in the system. This step exemplifies the precision and foresight required in algebraic problem-solving, where a single, well-executed multiplication can significantly simplify the path to the solution.
Step 2: Eliminating 'x'
Now we have our two equations:
- 3x - 2y = 6
- -3x + 12y = 24
Notice that the 'x' terms are now opposites. This is exactly what we wanted! With the 'x' terms perfectly aligned for elimination, the next step is to add the two equations together. This is where the magic of the elimination method truly shines, as the opposing terms cancel each other out, leaving us with a simpler equation. By carefully adding the left-hand sides and the right-hand sides separately, we maintain the balance of the equation while shedding one variable. This pivotal moment in the solution process transforms the problem from a complex system into a straightforward equation with a single unknown, paving the way for a clear and direct solution. The elegance of this step lies in its efficiency, streamlining the path to finding the values that satisfy the original system.
Let's add them together:
(3x - 2y) + (-3x + 12y) = 6 + 24
Simplifying, we get:
10y = 30
The beauty of the elimination method is on full display here. The 'x' terms vanished, leaving us with a simple equation involving only 'y'. The simplification process in the elimination method is a key step that transforms a complex equation into a manageable form. By combining like terms after eliminating a variable, we reduce the equation to its simplest expression, making it easier to isolate the remaining variable. This process not only streamlines the solution but also highlights the underlying elegance of algebra, where strategic manipulations can lead to significant simplifications. The result is a clear, concise equation that directly leads us to the value of one variable, bringing us closer to solving the entire system.
Step 3: Solving for 'y'
We have 10y = 30. To find 'y', we simply divide both sides by 10:
y = 30 / 10 y = 3
Great! We've found the value of 'y'. Now that we've successfully isolated y and found its value, we've cleared a major hurdle in solving the system of equations. This moment is significant because it marks the transition from a two-variable problem to a single-variable one, simplifying the remainder of the solution process. The clarity and precision of this step are hallmarks of the elimination method, demonstrating its effectiveness in breaking down complex problems into manageable parts. With the value of y in hand, we're now perfectly positioned to find the value of x, completing our journey to solve the system.
Step 4: Solving for 'x'
Now that we know y = 3, we can substitute this value into either of our original equations to solve for 'x'. Let's use the second equation, x - 4y = -8:
x - 4(3) = -8 x - 12 = -8
Add 12 to both sides:
x = -8 + 12 x = 4
Fantastic! We've found the value of 'x'. Substituting the value of y back into one of the original equations is a crucial step in completing the solution process. This technique allows us to leverage our newfound knowledge of one variable to solve for the other, effectively unraveling the system of equations. The flexibility to choose either equation adds to the method's efficiency, allowing us to select the one that simplifies the calculation. This step underscores the interconnectedness of the variables in a system, showing how solving for one piece of the puzzle illuminates the rest. By finding the value of x, we bring our solution to its final form, ready to be presented with confidence.
Step 5: The Solution
We found that x = 4 and y = 3. So, the solution to our system of equations is (4, 3). The culmination of our efforts in the elimination method brings us to the solution, a precise point that satisfies all equations in the system. This final answer is more than just a pair of numbers; it's the intersection point of the lines represented by our equations, a harmonious meeting place on the coordinate plane. Presenting the solution as an ordered pair, (x, y), provides clarity and completeness, making it easy to understand and verify. This moment of discovery underscores the power of algebraic techniques to solve complex problems, turning abstract equations into concrete solutions.
To be absolutely sure, let's plug these values back into our original equations to check:
- 3(4) - 2(3) = 12 - 6 = 6 (Correct!)
- 4 - 4(3) = 4 - 12 = -8 (Correct!)
Our solution checks out! Always remember to verify your solutions, guys. It's a great way to catch any mistakes and build confidence in your answer. Verifying the solution by substituting the values back into the original equations is a hallmark of thorough mathematical practice. This crucial step serves as a check, ensuring that our calculated values satisfy all conditions of the system. The satisfaction of seeing both equations hold true with our solution reinforces the accuracy of our work and solidifies our understanding of the elimination method. This final validation is not just a formality; it's a testament to the precision and reliability of our algebraic journey.
Wrapping Up
And there you have it! We've successfully solved the system of equations 3x - 2y = 6 and x - 4y = -8 using the elimination method. Remember, the key is to manipulate the equations so that one variable cancels out when you add or subtract them. The elimination method is a powerful tool in the world of algebra, offering a systematic approach to solving systems of equations. Its strength lies in its ability to simplify complex problems by strategically eliminating variables, paving the way for clear and concise solutions. This method is not just a set of steps to follow; it's a way of thinking, encouraging problem-solvers to look for patterns and create opportunities for simplification. Mastering the elimination method opens doors to tackling a wide range of mathematical challenges, from basic algebra to advanced applications in science and engineering.
Practice makes perfect, so try solving more systems of equations using this method. The more you practice, the more comfortable and confident you'll become. Keep up the great work, guys, and happy problem-solving!
