Solving Linear Equations A Step By Step Guide
Hey there, math enthusiasts! Today, we're diving into the fascinating world of systems of linear equations. You know, those sets of equations that look like a secret code just waiting to be cracked. We'll tackle a specific example, but more importantly, we'll equip you with the tools and understanding to conquer any system of linear equations that comes your way. So, let's put on our detective hats and get started!
Understanding Systems of Linear Equations
Before we jump into the solution, let's break down what a system of linear equations actually is. Linear equations are simply equations that represent straight lines when graphed. A system of these equations is just a collection of two or more of these lines. The solution to the system is the point (or points) where these lines intersect. Think of it like finding the meeting place of two different paths. This intersection point satisfies both equations simultaneously. There are several methods to solve these systems, and we'll explore one of the most common and effective ones: the elimination method.
The Elimination Method: Our Secret Weapon
The elimination method, also known as the addition method, is a clever technique that allows us to eliminate one variable by adding the equations together. The key is to manipulate the equations so that the coefficients of one variable are opposites. This way, when we add the equations, that variable disappears, leaving us with a single equation in one variable. This simplified equation is much easier to solve. Once we find the value of one variable, we can substitute it back into either of the original equations to find the value of the other variable. It's like a domino effect – solve for one, and the other falls into place. Let's see how this works in practice with our example.
Solving the System: A Step-by-Step Approach
Our mission, should we choose to accept it, is to solve the following system of linear equations:
y - 4x = 7
2y + 4x = 2
Notice anything special about these equations? Take a close look at the coefficients of the x terms. We have -4x in the first equation and +4x in the second equation. Bingo! They are opposites. This is exactly what we need for the elimination method to work its magic. This makes our work easier than if we had to change any of the equations before adding. Let's dive into each step to see exactly how this method makes our system of equations simple to solve and how to select the correct answer.
Step 1: Adding the Equations
The beauty of the elimination method is its simplicity. Since the coefficients of x are already opposites, we can directly add the two equations together. When we add the left-hand sides, we get (y - 4x) + (2y + 4x). When we add the right-hand sides, we get 7 + 2. So, our new equation looks like this:
y - 4x + 2y + 4x = 7 + 2
Now, let's simplify. Notice that the -4x and +4x terms cancel each other out, leaving us with:
3y = 9
Ta-da! We've eliminated x and now have a simple equation in just one variable, y. This is a major victory in our quest to solve the system.
Step 2: Solving for y
Now that we have the equation 3y = 9, solving for y is a piece of cake. We simply divide both sides of the equation by 3:
3y / 3 = 9 / 3
This gives us:
y = 3
Fantastic! We've found the value of y. Now, we're halfway to solving the system. We know that the y-coordinate of the solution is 3.
Step 3: Substituting to Find x
With the value of y in hand, we can now substitute it back into either of the original equations to find the value of x. It doesn't matter which equation we choose; the result will be the same. Let's choose the first equation, y - 4x = 7, for this example.
We replace y with 3 in the equation:
3 - 4x = 7
Now, we need to isolate x. First, subtract 3 from both sides:
3 - 4x - 3 = 7 - 3
This simplifies to:
-4x = 4
Next, divide both sides by -4:
-4x / -4 = 4 / -4
This gives us:
x = -1
Excellent! We've found the value of x. The x-coordinate of the solution is -1.
Step 4: The Solution! Coordinate Pairs
We've done it! We've found the values of both x and y. The solution to the system of linear equations is the point where the two lines intersect. We express this solution as an ordered pair (x, y). In our case, x is -1 and y is 3. Therefore, the solution is:
(-1, 3)
This means that the lines represented by the two equations intersect at the point (-1, 3) on the coordinate plane. This ordered pair satisfies both equations, which is the ultimate test of a solution to a system of equations.
Checking Our Answer: The Final Verification
It's always a good idea to double-check our answer to make sure we haven't made any mistakes. To do this, we substitute the values of x and y back into both original equations and see if they hold true. Remember, a true solution will satisfy both equations simultaneously.
Checking in the First Equation
Our first equation is y - 4x = 7. Let's substitute x = -1 and y = 3:
3 - 4(-1) = 7
Simplify:
3 + 4 = 7
7 = 7
It checks out! The equation holds true. So far, so good.
Checking in the Second Equation
Our second equation is 2y + 4x = 2. Let's substitute x = -1 and y = 3:
2(3) + 4(-1) = 2
Simplify:
6 - 4 = 2
2 = 2
It checks out again! The equation holds true. This confirms that our solution is indeed correct.
Identifying the Correct Option
Now that we've confidently solved the system and verified our solution, we can look back at the multiple-choice options and identify the correct one. Our solution is (-1, 3). Looking at the options:
A. (3, 1) B. (1, 3) C. (3, -1) D. (-1, 3)
We can clearly see that option D, (-1, 3), matches our solution. Therefore, option D is the correct answer.
Key Takeaways and Further Exploration
We've successfully navigated the world of systems of linear equations and emerged victorious! We used the elimination method to solve a specific example, but the principles we've learned can be applied to a wide range of problems. Remember these key takeaways:
- Systems of linear equations represent the intersection of lines.
- The elimination method is a powerful tool for solving these systems.
- Always check your answer to ensure accuracy.
Beyond Elimination: Other Methods
While the elimination method is a fantastic tool, it's not the only way to solve systems of linear equations. There's also the substitution method, where you solve one equation for one variable and substitute that expression into the other equation. Another approach is graphing, where you plot the lines and visually identify their intersection point. Each method has its strengths and weaknesses, and the best method to use often depends on the specific system of equations you're dealing with.
Real-World Applications
Systems of linear equations aren't just abstract mathematical concepts; they have real-world applications in various fields. From economics and engineering to computer science and physics, these systems help us model and solve problems involving multiple variables and relationships. For example, they can be used to determine the optimal mix of products to manufacture, analyze electrical circuits, or model the trajectory of a projectile.
Practice Makes Perfect
The best way to master solving systems of linear equations is to practice. Work through different examples, try different methods, and don't be afraid to make mistakes. Each mistake is a learning opportunity. The more you practice, the more confident and proficient you'll become.
Conclusion: You've Got This!
Solving systems of linear equations might seem daunting at first, but with the right tools and a systematic approach, it becomes a manageable and even enjoyable challenge. We've walked through the elimination method step-by-step, emphasizing the importance of understanding the underlying concepts and verifying our solutions. So, the next time you encounter a system of linear equations, remember the strategies we've discussed, embrace the challenge, and confidently find the solution. You've got this!
Now, go forth and conquer those equations, guys! Keep practicing, keep exploring, and keep enjoying the beauty of mathematics. Remember, every problem solved is a step forward in your mathematical journey.
