Solving Systems: Which Linear Combination Reveals Solutions?

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Hey guys! Let's dive into a fun math problem today that involves systems of equations and how we can figure out how many solutions they have. We'll break down a scenario where Alvin is using a specific method to solve a system, and we need to determine the best way to figure out the number of solutions. Ready to get started?

Understanding the Problem: Alvin's Approach

Let's set the stage. Imagine Alvin is faced with a system of two linear equations. These equations might look something like this:

  • Equation 1: Ax + By = C
  • Equation 2: Dx + Ey = F

Where A, B, C, D, E, and F are just numbers. Alvin's first move is to multiply the first equation by 2 and the second equation by -3. This is a common technique used in solving systems of equations called the elimination method. The goal here is to manipulate the equations so that when you add them together, one of the variables (either x or y) cancels out. This makes it easier to solve for the remaining variable.

So, after Alvin's multiplication, our equations now look like this:

  • New Equation 1: 2(Ax + By) = 2C => 2Ax + 2By = 2C
  • New Equation 2: -3(Dx + Ey) = -3F => -3Dx - 3Ey = -3F

Now, the crucial question is: Which linear combination of these modified equations will reveal the number of solutions to the system? To answer this, we need to understand what the different solution scenarios look like in the context of linear combinations.

Delving Deeper: Linear Combinations and Solutions

A linear combination simply means adding the two equations together. We're looking for a specific outcome from this addition that tells us about the number of solutions. Let's consider the possibilities:

  • One Unique Solution: This is the most common scenario. The two lines represented by the equations intersect at a single point. This point's coordinates (x, y) are the solution to the system.
  • No Solution: The lines are parallel and never intersect. This means there's no pair of (x, y) values that satisfy both equations simultaneously.
  • Infinitely Many Solutions: The lines are the same line! They overlap completely, so every point on the line is a solution. This happens when one equation is a multiple of the other.

So, how do these scenarios manifest when we add the equations together? This is where the magic happens.

Unpacking the Scenarios: Finding the Number of Solutions

To figure out the right linear combination, let's break down each solution scenario and see what the resulting equation looks like after adding the modified equations.

Scenario 1: One Unique Solution

In this case, when we add the equations, we won't get a trivial equation like 0 = 0. Instead, we'll get an equation that we can solve for one of the variables. For example, if the y terms cancel out, we'll be left with an equation in terms of x, which we can then solve. Once we find the value of x, we can plug it back into either of the original equations to find y. This gives us our single, unique solution.

Key Takeaway: If adding the equations results in an equation that can be solved for a variable, the system has one unique solution.

Scenario 2: No Solution

This is where things get interesting. If the system has no solution (parallel lines), when we add the modified equations, both the x and y terms will cancel out. But, the constant terms will not cancel out. This will result in a contradictory equation like 0 = 5 (or any other non-zero number). This contradiction tells us that there's no solution – the lines never intersect.

Key Takeaway: If adding the equations results in a contradiction (e.g., 0 = a non-zero number), the system has no solution.

Scenario 3: Infinitely Many Solutions

In this scenario, the lines are the same. When we add the modified equations, everything cancels out. Both the x and y terms, and the constant terms, all become zero. This results in the equation 0 = 0. This true but unhelpful equation tells us that the two original equations are essentially the same, and there are infinitely many solutions.

Key Takeaway: If adding the equations results in the identity 0 = 0, the system has infinitely many solutions.

The Linear Combination That Reveals All

Okay, so we've analyzed the scenarios. Now, let's bring it all together. The linear combination that reveals the number of solutions is simply adding the two modified equations together:

(2Ax + 2By) + (-3Dx - 3Ey) = 2C + (-3F)

Let's call this the Combined Equation. The form of this Combined Equation will tell us everything we need to know:

  1. If the Combined Equation can be solved for x or y: One unique solution.
  2. If the Combined Equation is a contradiction (0 = non-zero): No solution.
  3. If the Combined Equation is the identity (0 = 0): Infinitely many solutions.

So, by performing this linear combination, we've created a powerful tool for analyzing the system.

Putting It Into Practice: Examples and Tips

Let's look at a couple of quick examples to solidify this concept:

Example 1:

Suppose after Alvin's multiplication and addition, the Combined Equation is:

0 = 0

This tells us immediately that the system has infinitely many solutions.

Example 2:

Suppose the Combined Equation is:

5x = 10

This equation can be solved for x (x = 2). Therefore, the system has one unique solution.

Example 3:

Suppose the Combined Equation is:

0 = 7

This is a contradiction, so the system has no solution.

Pro Tips for Success

  • Pay close attention to signs: A small mistake with a plus or minus can completely change the outcome.
  • Double-check your work: It's easy to make arithmetic errors, especially when dealing with multiple steps.
  • Think about what you're trying to achieve: Remember, the goal is to eliminate one variable or identify a contradiction or identity.

Wrapping Up: The Power of Linear Combinations

So, there you have it! By understanding linear combinations and how they relate to the solutions of a system of equations, we can quickly determine the number of solutions. Alvin's approach of multiplying the equations by constants is a clever way to set up the problem, and by adding the resulting equations, we unlock the key to understanding the solution landscape.

Remember, math isn't just about crunching numbers; it's about understanding the underlying concepts and using them to solve problems. Keep practicing, and you'll become a system-solving pro in no time! Keep an eye out for more math adventures coming soon. Happy problem-solving, guys! 🚀