Hey guys! Let's dive into a cool math problem today. We're going to explore a system of equations that, interestingly, has no solution. This means the lines represented by these equations never intersect. We'll figure out why and then look at how we can create new equations that are linear combinations of the original ones.
Understanding the Problem
First, let's take a look at the system of equations we're dealing with:
\begin{cases}
\frac{2}{3} x + \frac{5}{2} y = 15 \\
4x + 15y = 12
\end{cases}
Systems of equations are sets of two or more equations containing the same variables. The solution to a system of equations is the set of values for the variables that make all the equations true simultaneously. Graphically, this means the point where the lines intersect. However, in our case, this system has no solution, indicating that these lines are parallel and never meet. We need to figure out why and how we can express this relationship mathematically.
Checking for Parallel Lines
The key to understanding why this system has no solution lies in the relationship between the coefficients of x and y in both equations. If the ratios of the coefficients of x and y are equal, but the ratios of the constants are different, the lines are parallel. Let's examine this:
In the first equation, we have . In the second equation, we have . To easily compare, let’s multiply the first equation by a constant to see if we can make the coefficients of x or y match those in the second equation. If we multiply the first equation by 6, we get:
Now, we can clearly see the equations in a new light:
\begin{cases}
4x + 15y = 90 \\
4x + 15y = 12
\end{cases}
Notice that the coefficients of x and y are the same (4 and 15, respectively), but the constants are different (90 and 12). This confirms that the lines are parallel because they have the same slope but different y-intercepts. Parallel lines, by definition, never intersect, so the system has no solution. Understanding this fundamental concept is crucial before we move on to linear combinations.
The Concept of Linear Combinations
Before we dive deeper, let’s make sure we’re all on the same page about what a linear combination actually means. A linear combination of equations is simply a new equation formed by adding or subtracting multiples of the original equations. Basically, you’re taking your original equations and playing around with them – multiplying them by numbers (constants) and then adding them together. The goal? To create a new equation that still holds true within the system.
For example, imagine you have two equations, let’s call them Equation A and Equation B. A linear combination could look like this: 2 * Equation A + 3 * Equation B. What we're doing here is multiplying Equation A by 2, multiplying Equation B by 3, and then adding the resulting equations together. This might seem a bit abstract now, but it's a powerful tool for solving systems of equations, especially when you want to eliminate variables or find new relationships between them.
In the context of our problem, where the system has no solution, linear combinations might not help us find a specific x and y that works for both equations. However, they can help us understand the relationship between the equations and how they lead to this “no solution” scenario. For instance, we can use linear combinations to show that the equations contradict each other, further solidifying why there’s no solution. We’ll explore this in more detail as we proceed.
Finding a Linear Combination
The question asks us which equation could represent a linear combination of the given system. This means we need to find an equation that can be obtained by multiplying the original equations by constants and then adding them together. This is where the fun begins, guys! We're going to play equation detectives, searching for the right combination.
Manipulating the Equations
Let's rewrite our original equations for clarity:
\begin{cases}
\frac{2}{3} x + \frac{5}{2} y = 15 \quad (1) \\
4x + 15y = 12 \quad (2)
\end{cases}
Our goal is to manipulate these equations to create a new equation that demonstrates the inconsistency of the system. Remember, a linear combination involves multiplying each equation by a constant and then adding the results. A clever approach here is to eliminate one of the variables (either x or y) to see what kind of equation we end up with. If we end up with a contradiction (like 0 = some non-zero number), it further proves that the system has no solution.
Let's try to eliminate x. To do this, we need to make the coefficients of x in both equations opposites of each other. We can multiply equation (1) by -6 to get a -4x term:
Now, we have a new equation (3) that we can combine with equation (2). Notice how the -4x in equation (3) is the opposite of the 4x in equation (2). This sets us up perfectly to eliminate x. Hang tight, guys, we’re getting closer to cracking this!
Adding the Equations
Now that we have
\begin{cases}
-4x - 15y = -90 \quad (3) \\
4x + 15y = 12 \quad (2)
\end{cases}
we can add equations (2) and (3) together. When we add these equations, the x terms cancel out (-4x + 4x = 0), and the y terms also cancel out (-15y + 15y = 0). This leaves us with:
Boom! We've hit the jackpot! The equation 0 = -78 is a clear contradiction. Zero cannot equal -78. This contradiction arises because the original system of equations has no solution. The lines they represent are parallel and never intersect, leading to this impossible situation. This result perfectly illustrates why linear combinations are so powerful. They can reveal hidden inconsistencies and provide deeper insights into the relationships between equations.
Interpreting the Result
The equation 0 = -78 is a linear combination of the original system, specifically obtained by multiplying the first equation by -6 and then adding it to the second equation. This resulting contradiction is the nail in the coffin for any potential solutions. It definitively proves that no values of x and y can satisfy both original equations simultaneously.
This also highlights a key concept in linear algebra: when a system of equations leads to a contradiction like this (0 = a non-zero number), the system is said to be inconsistent. Inconsistent systems have no solutions, and the equations within them are essentially fighting against each other, making it impossible to find a common ground.
Identifying the Correct Equation
Now, let's circle back to the original question: "Which equation could represent a linear combination of the system?" We’ve actually done the hard work already! We know that a valid linear combination leads to the contradiction 0 = -78. However, the options provided might not directly state “0 = -78”. Instead, they might present an equation that is equivalent to this contradiction or was a step in the process of getting the contradiction.
Analyzing Possible Answers
We need to look for an equation that reflects the relationship we found when we added the modified equations. Remember, we multiplied the first equation by -6 and added it to the second equation. This process eliminated both x and y, leaving us with the contradiction.
Let's say one of the options is:
This equation is not the direct contradiction we found (0 = -78). However, it might be an intermediate step in the process. To check, we’d need to see if this equation was generated during our linear combination process. Did we encounter this equation when we multiplied the first equation by -6? Let’s look:
Nope, this doesn't match our example option. So, this option is likely not the correct linear combination.
Finding the Equivalent Form
The correct answer will likely be an equation that, when simplified or rearranged, leads to the contradiction 0 = -78 or is a direct result of our linear combination process. It might also be a multiple of the contradiction. For example, 0 = -156 (which is 2 times -78) would also indicate the system has no solution.
To identify the correct equation, we need to carefully compare each option with the steps we took to arrive at our contradiction. We're essentially looking for a mathematical "fingerprint" – an equation that carries the mark of our linear combination process.
Conclusion
So, there you have it, guys! We've navigated through a system of equations with no solution, understood why it has no solution (parallel lines!), and explored the power of linear combinations. We've also learned how to identify a linear combination that represents the inconsistency of the system. Remember, the key is to manipulate the equations in a way that reveals the underlying contradiction. By multiplying the first equation by -6 and adding it to the second, we landed on the equation 0 = -78, a clear sign that no solution exists.
Linear combinations are a fantastic tool in the world of algebra, and understanding them can unlock many doors to solving complex problems. Keep practicing, and you'll become equation-solving pros in no time!