System Of Equations: Solutions And Dependency

Hey guys! Welcome back to Plastik Magazine, where we dive deep into all things awesome, including how to crack the code on some tricky math problems. Today, we're tackling a classic: systems of linear equations. You know, those pairs of equations that look like they're best buds, hanging out together. We've got a specific system here:

$\begin{array}{l} 8 x+2 y=8 \ y=-4 x+4 \end{array}$

Our mission, should we choose to accept it (and we totally will!), is to figure out how many solutions this system has. And, just as importantly, we need to determine if this system is dependent or independent. Sounds like a puzzle, right? Well, grab your thinking caps, because we're about to solve it!

Understanding Systems of Equations

Before we jump into our specific problem, let's get our heads around what a system of linear equations actually is. Think of each equation as a line on a graph. When you have a system, you're essentially looking at where these lines intersect. The solution(s) to the system are the points (or point!) where these lines meet. So, if they cross at one spot, you have one solution. If they never cross, no solutions. And if they're the exact same line (crazy, right?), then they have infinitely many solutions.

Now, about dependent vs. independent systems. An independent system is like two distinct lines that have a unique meeting point. They're related by the problem, but they aren't just two ways of saying the same thing. On the other hand, a dependent system is where the two equations actually represent the same line. It's like having two different ways of describing your favorite hangout spot – it's still the same spot! This means every point on that line is a solution, leading to infinite solutions.

So, our goal is to see where our two lines, $8x + 2y = 8$ and $y = -4x + 4$, decide to hang out. Do they meet once? Never? Or are they practically twins?

Method 1: Substitution - The Sneaky Approach

One of the coolest ways to solve these systems is through substitution. It's like a secret agent mission where you swap out a variable for its equivalent expression. In our system, the second equation, $y = -4x + 4$, is already giving us a nice, isolated 'y'. This is our golden ticket! We can take this entire expression for 'y' and substitute it into the first equation wherever we see 'y'.

Let's do it! Our first equation is $8x + 2y = 8$. We're going to replace that 'y' with $(-4x + 4)$. So, it becomes:

$8x + 2(-4x + 4) = 8$

Now, we just need to channel our inner algebra wizards and simplify this equation. First, distribute the 2:

$8x - 8x + 8 = 8$

Lookie here! The 'x' terms ($8x$ and $-8x$) cancel each other out, leaving us with:

$8 = 8$

Whoa! What does $8 = 8$ even mean? This isn't a value for 'x' or 'y'. This is a true statement. It means that no matter what 'x' and 'y' are, as long as they satisfy the second equation, they will always satisfy the first one too. This is the big clue, guys! When you do substitution (or elimination, which we'll get to) and end up with a true statement like $8 = 8$ or $0 = 0$, it signifies that the two equations are essentially the same line. They're just written differently.

This leads us to a crucial conclusion: because we got a true statement, this system has infinitely many solutions. Every point on the line represented by $y = -4x + 4$ is a solution to the system. Think about it – if the lines are identical, they overlap everywhere!

Method 2: Graphing - The Visual Vibe

Sometimes, you just gotta see it to believe it, right? Graphing is our visual approach to solving systems of equations. It's like checking out the actual hangout spot of our lines.

Our first equation is $8x + 2y = 8$. To graph this, it's often easiest to get it into slope-intercept form, which is $y = mx + b$, where 'm' is the slope and 'b' is the y-intercept. Let's rearrange:

$8x + 2y = 8$ Subtract $8x$ from both sides: $2y = -8x + 8$ Divide everything by 2: $y = -4x + 4$

Now, let's look at our second equation: $y = -4x + 4$.

Wait a second... they're exactly the same equation! This is the ultimate confirmation. When you graph two identical equations, you're not just graphing one line; you're graphing the same line twice. It's like drawing over the same path multiple times. The lines completely overlap each other.

So, what does this mean for solutions? Since the lines are perfectly aligned, they intersect at every single point along the line. That means there are infinitely many solutions for this system. Every coordinate pair $(x, y)$ that satisfies $y = -4x + 4$ is a valid solution to the system.

This visual confirmation solidifies what we found with substitution. The graphical method is super helpful for visualizing the relationship between the lines and understanding the nature of the solutions.

Method 3: Elimination - The Direct Confrontation

Let's bring in another heavyweight: elimination. This method is all about lining up the equations and either adding or subtracting them to make one of the variables disappear. It's a bit more direct than substitution, like going straight for the bullseye.

Our system is:

1. $8x + 2y = 8$
2. $y = -4x + 4$

To use elimination effectively, we usually want both equations in the standard form, $Ax + By = C$. The first equation is already there. Let's rearrange the second equation to match:

$y = -4x + 4$ Add $4x$ to both sides: $4x + y = 4$

Now, let's line them up:

$8x + 2y = 8$ $4x + y = 4$

To eliminate a variable, we need the coefficients of either 'x' or 'y' to be opposites. We could multiply the second equation by -2 to make the 'y' coefficients opposites (2y and -2y):

$-2(4x + y) = -2(4)$ $-8x - 2y = -8$

Now, let's add this modified second equation to the first equation:

$(8x + 2y) + (-8x - 2y) = 8 + (-8)$ $8x + 2y - 8x - 2y = 0$

Again, look at this! The 'x' terms ($8x$ and $-8x$) cancel out, and the 'y' terms ($2y$ and $-2y$) also cancel out. We're left with:

$0 = 0$

This is another true statement! Just like with substitution, when elimination leads to a true statement like $0 = 0$, it means the equations are dependent. They represent the same line. Therefore, this system has infinitely many solutions.

Elimination is a powerful tool, and seeing $0 = 0$ pop out confirms that our lines are, in fact, one and the same.

Dependent or Independent? The Final Verdict

We've used three different methods – substitution, graphing, and elimination – and all of them have led us to the same conclusion: the statement $8=8$ or $0=0$ (which are true statements) appeared. This is the universal sign for a dependent system.

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