Infinite Solutions: Find The Matching Linear Equation

Hey Plastik Magazine readers! Let's dive into a cool math problem today that's all about systems of linear equations. We're going to explore what it means for a system to have an infinite number of solutions and how to spot the equation that makes it happen. So, buckle up, math enthusiasts, and let's get started!

Understanding Infinite Solutions

Before we jump into the problem, let's quickly recap what it means for a system of linear equations to have an infinite number of solutions. Imagine two lines graphed on the same coordinate plane. If these lines perfectly overlap each other, they essentially become the same line. This means every single point on one line is also on the other, leading to an infinite number of intersection points – hence, infinite solutions.

In algebraic terms, this happens when two equations are just multiples of each other. Think of it like this: if you can multiply one equation by a constant and get the other equation, you've got yourself a system with infinite solutions. Identifying these equations is key to solving this type of problem. Remember, guys, the goal here is to find an equation that, while perhaps looking different on the surface, is fundamentally the same line as the given equation. This often involves looking for proportional relationships between the coefficients of x, y, and the constant terms. So, keep your eyes peeled for these multiples – they're your ticket to spotting those infinite solutions!

The Problem: Muriel's Equations

Okay, so here's the setup: Muriel's got a system of two linear equations, and she knows it has an infinite number of solutions. One of her equations is already revealed to us: 3y = 2x - 9. The challenge now is to figure out which of the given options could be the other equation in her system. To do this, we need to identify which equation is essentially a disguised version of 3y = 2x - 9. This means we're hunting for an equation that's a multiple of the original, ensuring that the two lines overlap perfectly on a graph.

The options we have are:

A. 2y = x - 4.5

B. 6y = 6x - 27

C. y = (3/2)x - 4.5

Let's roll up our sleeves and dive into each option to see which one fits the bill. Remember, our main goal here is to find an equation that, when simplified or manipulated, looks exactly like our starting equation or a simple multiple of it. This is where our algebraic skills come into play, guys! We'll need to use techniques like multiplying both sides of the equation by a constant, or rearranging terms, to see if we can make a match. So, let's get to it and find that hidden equation!

Analyzing the Options

Let's break down each option and see if it creates a system with infinite solutions when paired with 3y = 2x - 9.

Option A: 2y = x - 4.5

To determine if this equation results in infinite solutions, we need to see if it's a multiple of our original equation, 3y = 2x - 9. Let's try to manipulate the given equation to match the form of our original equation. We can start by isolating 'y' in both equations to make comparisons easier.

For the original equation, 3y = 2x - 9, we divide both sides by 3, which gives us y = (2/3)x - 3. Now, let's do the same for Option A: 2y = x - 4.5. Divide both sides by 2 to isolate 'y': y = (1/2)x - 2.25.

Comparing the two equations, y = (2/3)x - 3 and y = (1/2)x - 2.25, we see that the coefficients of 'x' (2/3 and 1/2) and the constant terms (-3 and -2.25) are not proportional. This indicates that the two lines have different slopes and y-intercepts, meaning they will intersect at exactly one point. Therefore, Option A does not create a system with infinite solutions.

Option B: 6y = 4x - 18

Now, let's examine Option B: 6y = 4x - 18. Our goal is still to determine if this equation is a multiple of 3y = 2x - 9. One way to approach this is to try and transform Option B into our original equation. If we can do that, we know we've found a match.

Looking at 6y = 4x - 18, we can see that each term is a multiple of 2. So, let's divide the entire equation by 2 to simplify it: (6y)/2 = (4x)/2 - 18/2, which simplifies to 3y = 2x - 9.

Wait a second! That's exactly our original equation! This means that Option B, 6y = 4x - 18, is essentially the same line as 3y = 2x - 9. When graphed, they would perfectly overlap, leading to an infinite number of intersection points. Therefore, Option B does create a system with infinite solutions. This is likely our answer, but let's check the other options just to be sure.

Option C: y = (3/2)x - 4.5

Let's tackle Option C: y = (3/2)x - 4.5. As before, we want to see if this equation is a multiple of our original, 3y = 2x - 9. To make the comparison easier, let's rewrite the original equation in slope-intercept form (y = mx + b), which is the same form as Option C. We already did this when analyzing Option A, resulting in y = (2/3)x - 3.

Now we can directly compare y = (2/3)x - 3 with y = (3/2)x - 4.5. Notice that the slopes (the coefficients of 'x') are (2/3) and (3/2), which are reciprocals of each other. This means the lines are not parallel, and they will intersect at a single point. Also, the y-intercepts (-3 and -4.5) are different. Thus, Option C does not create a system with infinite solutions.

The Solution

After analyzing all the options, we've pinpointed the equation that creates a system with infinite solutions. Remember, we were looking for an equation that, when paired with 3y = 2x - 9, would represent the same line. And the winner is...

Option B: 6y = 4x - 18

We found that 6y = 4x - 18 is simply a multiple of 3y = 2x - 9 (we divided by 2 to show this). This means that these two equations represent the exact same line on a graph, leading to an infinite number of solutions. So, there you have it, guys! We successfully navigated this system of equations problem. Remember, the key to solving these types of questions is to recognize when equations are multiples of each other – that's your signal for infinite solutions.

Key Takeaways

Let's recap the main points we've covered in this problem. Understanding these key concepts will help you tackle similar questions with confidence.

• Infinite Solutions Mean Overlapping Lines: A system of linear equations has infinite solutions when the equations represent the same line. Graphically, this means the lines overlap completely, sharing every point.
• Multiples are the Key: To identify equations that represent the same line, look for multiples. If one equation can be obtained by multiplying the other equation by a constant, they are essentially the same line.
• Transforming Equations: Manipulating equations into the same form (like slope-intercept form, y = mx + b) makes it easier to compare them and identify if they are multiples of each other.

By keeping these takeaways in mind, you'll be well-equipped to handle any problem involving systems of linear equations and infinite solutions. Keep practicing, guys, and you'll become math masters in no time!

Wrapping Up

So, that's a wrap on our math adventure for today! We've successfully tackled a problem involving systems of linear equations and infinite solutions. Remember, the key takeaway is that infinite solutions occur when the equations represent the same line, and we can identify these by looking for multiples. This kind of problem shows how important it is to understand the underlying concepts in math. It's not just about plugging in numbers; it's about understanding what the equations represent and how they relate to each other.

We hope you found this exploration helpful and maybe even a little bit fun. Math can be like a puzzle, and it's super satisfying when you figure out the solution. Keep honing your skills, stay curious, and remember that every problem is an opportunity to learn something new. Until next time, keep those mathematical gears turning, guys! And don't forget to check back for more exciting math explorations here at Plastik Magazine. There's always something new to discover!