Solving Systems Of Equations: A Step-by-Step Guide

Hey everyone! Today, we're diving into the world of solving systems of equations algebraically. This might sound a bit intimidating, but trust me, it's totally manageable. We'll break down the process step-by-step, making sure you understand how to tackle these problems like a pro. Systems of equations pop up everywhere in math, so mastering this skill is super important. We'll focus on the specific problem you gave, but the techniques we use can be applied to many different scenarios. We'll use the two equations, $4x - 2y = 4$ and $6x - 4y = 6$, and find out whether there is one solution, no solution, or infinitely many solutions. Let's get started!

Understanding Systems of Equations and the Goal

First things first, what exactly is a system of equations? Basically, it's a set of two or more equations that we need to solve together. The solution to a system of equations is the set of values for the variables (in this case, x and y) that make all the equations true simultaneously. Think of it like finding a common ground – a point that satisfies all the equations involved. There are a few possibilities when solving these systems: a unique solution (the lines intersect at one point), no solution (the lines are parallel and never intersect), or infinitely many solutions (the lines are the same, overlapping each other). Our goal is to figure out which of these scenarios applies to the equations we were given. For the system to have a unique solution, the lines would intersect at one point. If the lines are parallel, there will be no solution. If the lines are coincident, which means they are the same line, then we have infinitely many solutions. This means there are infinite points in the equation that satisfy the system. Remember, the solution to a system is a set of values (x, y) that satisfy all the equations. Let's solve the equations!

Now, there are several methods to solve these systems, and each has its own advantages. We can use the substitution method, the elimination method, or even graphing. For this example, let’s go with the elimination method, as it's often the most straightforward approach for this type of problem. The elimination method involves manipulating the equations to eliminate one of the variables, making it easier to solve for the other. Basically, we want to add or subtract the equations in a way that cancels out either x or y. Then, we can solve for the remaining variable and use that value to solve for the variable we had previously eliminated. Let's see how it works in practice.

Step-by-Step: Solving the System Using Elimination

Alright, let’s get down to the nitty-gritty and solve our system of equations: $4x - 2y = 4$ and $6x - 4y = 6$. Here's how we'll do it using the elimination method:

Step 1: Manipulate the Equations

Our first aim is to make the coefficients of either x or y the same (or additive inverses – meaning they add up to zero) in both equations. Looking at our equations: $4x - 2y = 4$ and $6x - 4y = 6$, we can work with the y terms. We can multiply the first equation by -2 to make the y terms additive inverses. Doing this to the first equation gives us:

-2 * (4x - 2y = 4) => -8x + 4y = -8

Now, our system looks like this:

• -8x + 4y = -8
• 6x - 4y = 6

Step 2: Eliminate a Variable

Now we add the two equations together. Notice how the y terms have opposite signs and the same coefficient (+4y and -4y). This means they will cancel each other out when we add the equations:

(-8x + 4y) + (6x - 4y) = -8 + 6

This simplifies to:

-2x = -2

Step 3: Solve for the Remaining Variable

We're left with a single-variable equation: -2x = -2. Solving for x is easy, we divide both sides by -2:

x = -2 / -2

x = 1

So, we have found that x = 1. But we are not done yet, we still need to find y.

Step 4: Substitute and Solve for the Other Variable

Now that we know the value of x, we can substitute it into either of the original equations to solve for y. Let's use the first equation: $4x - 2y = 4$. Substitute x = 1:

4(1) - 2y = 4

4 - 2y = 4

Subtract 4 from both sides:

-2y = 0

Divide both sides by -2:

y = 0

So, we've found that y = 0. Therefore, our solution is the point (1, 0).

Step 5: Verify the Solution

It's always a good idea to verify your solution by plugging the values of x and y back into both of the original equations to make sure they work. Let's check: Original equation 1: $4x - 2y = 4$ Substitute x = 1 and y = 0: 4(1) - 2(0) = 4, so 4 - 0 = 4. This is correct! Original equation 2: $6x - 4y = 6$ Substitute x = 1 and y = 0: 6(1) - 4(0) = 6, so 6 - 0 = 6. This is also correct! Since the solution (1, 0) satisfies both equations, we know our answer is correct.

Analyzing the Solution and Understanding the Options

Now that we've found our solution, (1, 0), let's look back at the options given to us:

A. no solution B. many solutions C. (0,1) D. (1,0)

We found that the solution to our system of equations is (1, 0). Thus, the correct answer is D. This means the two lines represented by the equations intersect at the point (1, 0).

The Geometric Interpretation

It's helpful to visualize what's going on here. Each equation in a system of equations represents a straight line when graphed. The solution to the system is the point where the lines intersect. If there is no solution, it means the lines are parallel and never intersect. If there are many solutions, it means the lines are the same (they overlap each other). In our case, the lines intersect at (1, 0). Let's take a look at the graph! The graph of the equation $4x - 2y = 4$ is a straight line. The graph of the equation $6x - 4y = 6$ is also a straight line. Both of these lines intersect at the point (1,0). This point is the solution to the system of equations. The lines cross at the point (1, 0) and the two lines are not parallel and not the same. This also means we don't have no solution or many solutions.

Conclusion: Mastering the Algebra of Equations

So there you have it! We've successfully solved our system of equations using the elimination method. We found the solution to be (1, 0), which means the two lines intersect at that point. By following these steps and understanding the underlying concepts, you can confidently solve any system of equations thrown your way! Remember, practice makes perfect. Try working through more examples on your own. Keep an eye out for different types of equations and how the different methods might apply. Keep practicing, and you'll be an expert in no time! Keep in mind that you can check the solution by plugging the values of x and y back into the original equations. This is a very important step to check the validity of your answer. That's all for today, and I hope this helped you better understand how to solve systems of equations. Until next time, keep exploring the awesome world of math, and keep solving! Feel free to ask any questions in the comments below. Happy solving, guys!