Systems Of Equations: Finding The Number Of Solutions

Hey math whizzes! Ever stared at a system of equations and wondered, "How many answers are we gonna get here?" Well, you've come to the right place, guys. Today, we're diving deep into the world of linear systems and figuring out exactly how many solutions they have, and what that means for us. We'll be tackling a classic example:

$y=x-3$ $y=-4 x+3$

This little problem is super common in algebra, and understanding it is key to unlocking more complex math concepts. We're going to break down the process step-by-step, explaining why we get the answers we do. So, grab your notebooks, maybe a snack, and let's get this math party started! We'll be looking at whether the system is consistent or inconsistent, and if it's dependent or independent. These terms might sound a bit fancy, but they're actually pretty straightforward once you get the hang of them. We're aiming to figure out if we have one solution, infinite solutions, or no solution at all. Let's get this problem solved!

Understanding the Types of Solutions

Alright, let's get down to the nitty-gritty of what those terms – consistent, inconsistent, dependent, and independent – actually mean when we're talking about systems of equations. Think of it like this: a system of equations is like a conversation between two or more lines (or other shapes, but we're sticking to lines for now). The solutions are where those lines agree, or intersect.

Consistent systems are the ones that actually have at least one solution. This means the lines in our system are going to meet up somewhere. If they meet at exactly one point, that's your single solution. If they overlap perfectly (which means they are the same line), then they agree everywhere, leading to infinite solutions.

Now, inconsistent systems are the total opposite. These systems have no solution. Imagine two parallel lines that never, ever cross. That's an inconsistent system. No matter how far you extend them, they'll never meet, so there's no point where they agree.

We also have subcategories within consistent systems: independent and dependent.

• An independent system has exactly one unique solution. This is like two different lines crossing at a single point. They have a specific meeting place, and that's it. Our example system, $y=x-3$ and $y=-4x+3$, looks like it might fall into this category because the slopes are different.
• A dependent system has infinitely many solutions. This happens when the two equations in your system actually represent the exact same line. So, every single point on that line is a solution because the lines are perfectly on top of each other. It's like saying the same thing in two different ways.

So, to recap:

• Consistent: Has at least one solution.
  • Independent: Exactly one solution.
  • Dependent: Infinitely many solutions.
• Inconsistent: No solution.

Knowing these definitions is super important because when we solve our system, the result will tell us which of these categories our specific problem falls into. We're going to use these terms to describe the answer to our initial problem, so keep them fresh in your mind. It's all about whether the lines meet, and if so, how many times they meet!

Solving the System: Substitution Method

Alright guys, let's actually solve the system we've got:

$y=x-3$ $y=-4 x+3$

There are a couple of popular ways to solve systems of linear equations, but one of the most common and effective is the substitution method. This method is awesome because it lets us take one equation and plug it into another, simplifying the whole thing down to one variable.

Since both of our equations are already solved for $y$, this makes substitution super easy. We can take the expression for $y$ from the first equation ($y=x-3$) and substitute it into the second equation wherever we see $y$. So, the second equation, $y = -4x + 3$, becomes:

$(x-3) = -4x + 3$

See what we did there? We replaced the '$y$' on the left side with '$x-3$'. Now we have an equation with only '$x$' in it, which is way easier to handle.

Our next step is to isolate '$x$'. Let's get all the '$x$' terms on one side and the constants on the other. First, I like to add $4x$ to both sides of the equation to get rid of the negative '$x$' term on the right:

$x - 3 + 4x = -4x + 3 + 4x$

This simplifies to:

$5x - 3 = 3$

Now, let's get the '-3' away from the '$5x$' by adding 3 to both sides:

$5x - 3 + 3 = 3 + 3$

Which gives us:

$5x = 6$

Finally, to get '$x$' all by itself, we divide both sides by 5:

$x = 6/5$

Awesome! We've found the value for $x$. The substitution method worked like a charm. Now that we have $x$, we can easily find $y$ by plugging this value back into either of our original equations. Let's use the first one because it looks a bit simpler:

$y = x - 3$

Substitute $x = 6/5$ into this equation:

$y = (6/5) - 3$

To subtract these, we need a common denominator. Since 3 is the same as $15/5$:

$y = 6/5 - 15/5$

$y = -9/5$

So, we've found our solution! The point where these two lines intersect is at $(6/5, -9/5)$. This single, specific point tells us that our system has exactly one solution. We successfully used the substitution method to crack this problem, and it wasn't so bad, right?

Analyzing the Solution: Consistent, Independent, and Unique

Now that we've gone through the math and found our solution, let's put it all together and answer the question completely. We found that the system:

$y=x-3$ $y=-4 x+3$

has a unique solution at the point $(6/5, -9/5)$. What does this tell us about the system itself?

Remember our definitions from earlier?

• A system is consistent if it has at least one solution. Since we found one solution, our system is definitely consistent. Hooray!
• A system is independent if it has exactly one unique solution. We found exactly one point $(6/5, -9/5)$ where these two lines intersect. This means the lines are distinct and cross at only one place. Therefore, our system is independent.

So, putting it all together, our system has 1 solution, and it is consistent and independent. This perfectly matches option (A) from the choices provided.

Why isn't it the other options?

• Infinite solutions (B & C): Infinite solutions happen when the two equations represent the exact same line. This occurs when the equations have the same slope and the same y-intercept. In our case, the slopes are different (1 and -4), so the lines are not the same, meaning we can't have infinite solutions. Also, for a system to be dependent (which leads to infinite solutions), the equations must be multiples of each other, which is clearly not the case here.
• No solution (D): No solution happens when the lines are parallel but distinct. Parallel lines have the same slope but different y-intercepts. Our lines have different slopes (1 and -4), so they are not parallel and will definitely intersect. Thus, we don't have 'no solution'.

Our specific slopes (1 and -4) being different is the key indicator that we'll have exactly one intersection point, leading to a consistent and independent system with one solution. It's like two roads heading in different directions – they'll cross paths at exactly one spot!

Alternative Method: Graphing

While substitution is a super reliable way to find the exact solution, sometimes just visualizing the system can give you a great idea of how many solutions you'll have. Let's think about graphing our two equations:

1. $y = x - 3$ This is a line with a slope of $m=1$ (meaning for every 1 unit you go right, you go 1 unit up) and a y-intercept of $b=-3$ (meaning it crosses the y-axis at the point (0, -3)).

2. $y = -4x + 3$ This is a line with a slope of $m=-4$ (meaning for every 1 unit you go right, you go 4 units down) and a y-intercept of $b=3$ (meaning it crosses the y-axis at the point (0, 3)).

Now, picture drawing these two lines on a graph. You'd start by plotting the y-intercepts. For the first line, you plot (0, -3). For the second, you plot (0, 3). Then, using the slopes, you'd draw the lines. The first line goes up and to the right, while the second line goes steeply down and to the right.

Because the slopes are different ($1 eq -4$), these two lines are guaranteed to intersect at exactly one point. They aren't parallel (same slope, different intercept), and they aren't the same line (same slope, same intercept). They are just two distinct lines with different directions.

When two lines on a graph have different slopes, they must cross somewhere. That single crossing point is the unique solution to the system. This visual understanding reinforces what we found algebraically: there is one solution. This also confirms that the system is consistent (because there's an intersection) and independent (because there's only one intersection). So, even without calculating the exact coordinates, just by looking at the slopes and intercepts, we can determine the nature of the solution!

Conclusion: One Solution, Consistent, and Independent!

So, after all that math and talking, we've firmly landed on our answer. For the system of equations:

$y=x-3$ $y=-4 x+3$

We discovered through the substitution method that there's a single intersection point at $(6/5, -9/5)$. This unique intersection means:

• There is 1 solution.
• The system is consistent because it has a solution.
• The system is independent because there is only one unique solution.

This aligns perfectly with option (A). Remember guys, understanding the terminology – consistent, inconsistent, dependent, and independent – is just as crucial as the calculation itself. It helps us describe the behavior of the lines represented by our equations. Keep practicing these concepts, and soon you'll be spotting the number of solutions like a pro! Happy solving!