Solving Linear Systems: How Many Solutions Exist?

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Hey guys! Let's dive into the fascinating world of linear equations and explore how to figure out the number of solutions a system might have. We'll tackle the system:

y=−12x+4x+2y=−8\begin{aligned}y & = -\frac{1}{2} x+4 \\x+2 y & = -8\end{aligned}

and break down the steps to determine whether it has one solution, no solution, or an infinite number of solutions. So, buckle up, and let's get started!

Understanding Linear Systems and Their Solutions

Before we jump into solving the given system, let's quickly recap what a system of linear equations is and what the possible solution scenarios are. A linear system is simply a set of two or more linear equations involving the same variables. Think of it as a collection of straight lines on a graph. Now, when we talk about solutions, we're essentially looking for the point(s) where these lines intersect. There are three possible scenarios:

  1. One Solution: The lines intersect at exactly one point. This means the system has a unique solution, represented by the coordinates (x, y) of that intersection point.
  2. No Solution: The lines are parallel and never intersect. In this case, the system is inconsistent and has no solution.
  3. Infinite Solutions: The lines are coincident, meaning they overlap completely. Every point on the line is a solution, leading to an infinite number of solutions.

Understanding these scenarios is crucial because it helps us interpret the results we get when solving a linear system. We're not just looking for numbers; we're trying to understand the relationship between the lines represented by the equations.

Now, how do we actually figure out which scenario applies to a given system? There are several methods, including graphing, substitution, and elimination. Each method has its strengths and weaknesses, but they all aim to achieve the same goal: to find the values of the variables that satisfy all equations in the system simultaneously. In the next section, we'll apply one of these methods to our specific system and see what we uncover. So, stay tuned as we unravel the mystery of how many solutions our system has!

Solving the System Using Substitution

Okay, let's roll up our sleeves and solve the system:

y=−12x+4x+2y=−8\begin{aligned}y & = -\frac{1}{2} x+4 \\x+2 y & = -8\end{aligned}

We're going to use the substitution method here, which is particularly handy when one of the equations is already solved for one variable (like our first equation, which is solved for y). The idea behind substitution is simple: we'll substitute the expression for y from the first equation into the second equation. This will give us a new equation with only one variable (x), which we can then solve.

So, let's take the expression for y from the first equation, which is y = -1/2x + 4, and plug it into the second equation:

x + 2y = -8

becomes

x + 2(-1/2x + 4) = -8

Now, we need to simplify and solve for x. First, distribute the 2:

x - x + 8 = -8

Notice anything interesting? The x terms cancel each other out!

8 = -8

This is a bit of a head-scratcher, isn't it? We've ended up with a statement that is clearly false: 8 is not equal to -8. This is a crucial clue that tells us something important about our system. Remember the three scenarios we talked about earlier? This situation points us to one of them in particular. Think about it: if the equations were to have a common solution, we would have arrived at a true statement. But because we ended up with a false statement, it means there's no solution that can satisfy both equations simultaneously. In the next section, we'll interpret this result in the context of our linear system and draw our final conclusion.

Interpreting the Result: No Solution

Alright, guys, we've arrived at a fascinating point in our solution process. We plugged the first equation into the second and ended up with the contradictory statement 8 = -8. This isn't just a random mathematical hiccup; it's a powerful indicator that our system of equations has no solution. But what does this really mean in terms of the lines represented by these equations?

Remember, each linear equation represents a straight line on a graph. When we solve a system of linear equations, we're essentially trying to find the point(s) where these lines intersect. If the lines intersect at one point, we have one solution. If they intersect at every point (i.e., they are the same line), we have infinite solutions. But if they never intersect, we have no solution. And that's precisely what's happening here.

The fact that we arrived at a false statement tells us that the two lines represented by our equations are parallel. Parallel lines, by definition, never intersect. They run alongside each other, maintaining a constant distance, but never meeting. This geometric interpretation helps us visualize why there's no solution: there's simply no point (x, y) that lies on both lines simultaneously.

To further confirm this, we could rewrite the second equation, x + 2y = -8, in slope-intercept form (y = mx + b) to compare its slope and y-intercept with the first equation. Let's do that:

2y = -x - 8 y = -1/2x - 4

Now, let's compare this to our first equation:

y = -1/2x + 4

Notice that both equations have the same slope (-1/2), but different y-intercepts (4 and -4). This confirms that the lines are indeed parallel and will never intersect. So, the mystery is solved! Our system of linear equations has no solution.

Final Answer and Key Takeaways

So, drumroll please... the answer to our question, "How many solutions does this linear system have?" is C. no solution. We arrived at this conclusion by using the substitution method and encountering a contradictory statement (8 = -8), which indicated that the lines represented by the equations are parallel and never intersect.

Let's recap the key takeaways from our journey:

  • Linear systems can have one solution, no solution, or an infinite number of solutions.
  • The substitution method is a powerful tool for solving systems of equations, especially when one equation is already solved for a variable.
  • A contradictory statement (like 8 = -8) when solving a system indicates that the system has no solution.
  • Lines with the same slope but different y-intercepts are parallel and will never intersect.

Understanding these concepts will help you tackle a wide range of linear system problems with confidence. Remember, the key is to carefully apply the methods, interpret the results, and visualize what's happening with the lines themselves. Keep practicing, and you'll become a linear system master in no time!