Solving System Of Equations: 2x-4y=0 And 3x-6y=2
Hey guys! Let's dive into solving a system of equations today. We've got two equations here: 2x - 4y = 0 and 3x - 6y = 2. Don't worry, it might look intimidating at first, but we'll break it down step by step. Understanding how to solve systems of equations is super useful in math and even in real-life situations where you need to find the values of multiple unknowns. So, let's get started!
Understanding Systems of Equations
Before we jump into solving, let's quickly chat about what a system of equations actually is. Essentially, it's a set of two or more equations that involve the same variables. Our goal? To find the values for those variables that make all the equations true at the same time. Think of it like finding the perfect puzzle pieces that fit together in multiple places. In our case, we're looking for the values of x and y that satisfy both 2x - 4y = 0 and 3x - 6y = 2.
Why is this important? Well, systems of equations pop up all over the place! From figuring out the cost of items when you only know the combined price, to modeling complex relationships in science and economics, these skills are incredibly valuable. So, by mastering this, you're not just doing math; you're building problem-solving muscles.
When tackling systems of equations, there are a few main methods we can use: substitution, elimination, and graphing. Each method has its strengths, and sometimes one method will be easier to use than another depending on the specific equations we're dealing with. Today, we'll primarily focus on using the substitution and elimination methods to crack our problem. We will explore each method in detail, making sure you understand not just the how, but also the why behind each step. This way, you'll be well-equipped to handle any system of equations that comes your way.
Method 1: The Substitution Method
The substitution method is all about isolating one variable in one equation and then plugging that expression into the other equation. It's like taking one piece of information and using it to unlock another. Let's see how this works with our equations:
- 2x - 4y = 0
- 3x - 6y = 2
Step 1: Isolate a Variable
Looking at equation 1 (2x - 4y = 0), it seems easier to isolate x. Let's add 4y to both sides:
2x = 4y
Now, divide both sides by 2:
x = 2y
Great! We've got x by itself. This is our key to substitution.
Step 2: Substitute
Now, we'll take this expression for x (x = 2y) and substitute it into equation 2 (3x - 6y = 2). This means replacing the x in equation 2 with "2y":
3(2y) - 6y = 2
See what we did there? We've now got an equation with just one variable, y. This is much easier to solve!
Step 3: Solve for y
Let's simplify and solve for y:
6y - 6y = 2
0 = 2
Wait a minute... 0 = 2? That's not right! This is a contradiction. It means there's no value of y that can make this equation true.
Step 4: Interpret the Result
Since we arrived at a contradiction (0 = 2), this tells us something important: the system of equations has no solution. This means there are no values for x and y that will satisfy both equations simultaneously.
But what does this mean graphically? Well, if we were to graph these two lines, they would be parallel. Parallel lines never intersect, which is why there's no solution – there's no point that lies on both lines.
Method 2: The Elimination Method
The elimination method (also sometimes called the addition method) is another powerful technique for solving systems of equations. The basic idea is to manipulate the equations so that when you add them together, one of the variables cancels out. It's like strategically combining the equations to eliminate a variable and simplify the problem.
Let's revisit our system:
- 2x - 4y = 0
- 3x - 6y = 2
Step 1: Manipulate the Equations
To eliminate a variable, we need to make the coefficients of either x or y opposites. Looking at our equations, let's focus on eliminating x. The coefficients of x are 2 and 3. The least common multiple of 2 and 3 is 6. So, we want to make one coefficient 6 and the other -6.
Let's multiply equation 1 by -3:
-3(2x - 4y) = -3(0)
-6x + 12y = 0
Now, let's multiply equation 2 by 2:
2(3x - 6y) = 2(2)
6x - 12y = 4
Now we have a new system:
- -6x + 12y = 0
- 6x - 12y = 4
Notice that the coefficients of x are now opposites (-6 and 6). Also, the coefficients of y are opposites (12 and -12). This sets us up perfectly for elimination!
Step 2: Add the Equations
Now, let's add the two equations together:
(-6x + 12y) + (6x - 12y) = 0 + 4
-6x + 6x + 12y - 12y = 4
0 = 4
Step 3: Interpret the Result
Just like with the substitution method, we've arrived at a contradiction: 0 = 4. This is not a true statement. Again, this tells us that the system of equations has no solution.
This confirms what we found using substitution. The equations represent parallel lines, and there is no point of intersection.
Why No Solution?
You might be wondering,
