Decoding $0=60$: No Solution With Linear Combination

Hey there, math enthusiasts and problem-solvers! Ever been working through a system of equations, feeling super confident, and then bam! you hit an equation like $0=60$? If you're anything like Heather, who recently encountered this exact scenario using the linear combination method, you might be scratching your head and wondering, "What on earth does $0=60$ even mean?" Well, don't sweat it, guys! This isn't a sign that the math universe has broken; it's actually a super important clue telling you something specific about the lines you're dealing with. Let's dive deep into Heather's situation and decode this mathematical mystery together. Understanding what happens when you get a contradictory statement like $0=60$ is key to mastering systems of equations, especially with the linear combination method, and it really shows you the power of algebraic problem-solving. This kind of outcome isn't an error in your calculation (assuming you did everything right, of course!), but rather a specific type of result that tells a compelling story about the relationship between two lines on a coordinate plane. So, grab your pens and paper, because we're about to unveil the secrets behind this seemingly puzzling equation!

Understanding Heather's System of Equations

Heather's system of equations is a classic example of two linear equations representing lines on a coordinate plane. She's got these two bad boys: $-6x + 18y = 0$ and $4x - 12y = 20$. Now, for those of you who might be new to this, a system of equations is essentially a fancy way of saying we have two (or more!) equations that we want to solve simultaneously. The goal is often to find a point $(x, y)$ that satisfies both equations at the same time. Geometrically speaking, this point is where the two lines intersect. Think of it like trying to find the exact spot where two roads cross paths on a map. When you're looking at a system like Heather's, you're trying to figure out if these two lines ever meet, and if so, where. There are a few cool ways to tackle these systems: you could graph them and see where they cross, use substitution to swap one variable out for an expression, or, as Heather did, employ the powerful linear combination method, also famously known as the elimination method. Each method has its strengths, but the linear combination method is often preferred for its elegance and efficiency, especially when dealing with equations set up like these. It's all about strategically manipulating the equations to make one of the variables disappear, making the whole thing much easier to solve. Heather's specific equations might look a bit intimidating at first, but trust me, they're perfectly manageable once you know the tricks of the trade. The first equation, $-6x + 18y = 0$, can be simplified or rearranged to show a clear relationship between $x$ and $y$. Similarly, $4x - 12y = 20$ is another linear equation, and together, they form a dynamic duo waiting to be solved. The key is to remember that we're always looking for a common ground, a point $(x, y)$ that makes both statements true. But what if there isn't one? That's where Heather's result of $0=60$ comes into play, signaling a very specific and important scenario in the world of linear algebra. We're about to see how this method can reveal not just solutions, but also the absence of solutions, and what that truly implies for our lines on the coordinate plane. Understanding the starting point, these two linear equations, is the crucial first step before we can appreciate the profound meaning of Heather's surprising result. It’s like setting the stage for a big reveal!

Diving Deep into the Linear Combination Method (Elimination)

Alright, let's get into the nitty-gritty of the linear combination method, which is often called the elimination method because, well, you're literally trying to eliminate a variable! This method is super effective when you want to solve a system of equations by adding or subtracting the equations to get rid of either the $x$ or $y$ variable. The trick is to manipulate one or both equations so that when you add them together, the coefficients of one variable are opposites (like $5x$ and $-5x$). Let's re-examine Heather's system:

1. $-6x + 18y = 0$
2. $4x - 12y = 20$

Now, to get to $0=60$, Heather must have performed a series of strategic multiplications and additions. Let's walk through how that might have happened. To eliminate a variable, you need to find a least common multiple for the coefficients of either $x$ or $y$. Let's consider eliminating $x$. The coefficients are $-6$ and $4$. The least common multiple of $6$ and $4$ is $12$. So, we want to make one $12x$ and the other $-12x$.

• To get $12x$ from $4x$, you'd multiply the second equation by $3$. So, $3 imes (4x - 12y = 20)$ becomes $12x - 36y = 60$.
• To get $-12x$ from $-6x$, you'd multiply the first equation by $2$. So, $2 imes (-6x + 18y = 0)$ becomes $-12x + 36y = 0$.

Now we have our new, modified system:

1. $-12x + 36y = 0$
2. $12x - 36y = 60$

This is where the magic (or the mystery!) happens. When you add these two equations together, term by term, watch what happens:

• $(-12x) + (12x) = 0x$
• $(36y) + (-36y) = 0y$
• $(0) + (60) = 60$

So, adding the left sides gives us $0x + 0y$, which is just $0$. And adding the right sides gives us $0 + 60$, which is $60$. Voila! We end up with the equation $0 = 60$. This is exactly how Heather arrived at her puzzling result! The linear combination method, in this case, didn't give her a value for $x$ or $y$; instead, it led to a clear contradiction. This isn't an error on Heather's part, but a critical piece of information revealed by the algebraic process. It shows the method working exactly as intended, highlighting a specific characteristic of the original system. The fact that both $x$ and $y$ terms vanished simultaneously, leaving only constants, is the key indicator here. If only one variable vanished, we'd be on our way to finding a solution. But when everything on the variable side disappears, and you're left with an untrue statement like $0 = 60$, it's screaming a very specific message about the relationship between those two lines. It’s like the equations themselves are telling you, “Hey, something’s up here!” It's a crucial point in understanding how systems of equations behave and what their various outcomes signify.

The Meaning Behind 0 = 60: Parallel Lines, No Solution!

Okay, so what's the big reveal behind that head-scratching $0=60$ equation? When you're using the linear combination method and both your variables (the $x$ and $y$ terms) vanish, leaving you with a statement that is mathematically impossible or a contradiction—like $0=60$, $5=10$, or any other situation where a number doesn't equal itself—it means one thing and one thing only: there is no solution to the system of equations! This isn't a math error, guys; it's a profound mathematical truth about the relationship between Heather's two lines. Geometrically, this result tells us that the two lines represented by Heather's equations are parallel and distinct. Think about it: if two lines are parallel, they run alongside each other forever without ever crossing. If they never intersect, there can't be a common point $(x, y)$ that satisfies both equations. Hence, no solution. It's like two railway tracks that go on endlessly, always side by side, but never meeting. They share the same direction but are never in the same place at the same time. The algebra perfectly reflects this geometric reality. To confirm this, we can always check the slopes of the original equations. Remember, parallel lines have the same slope but different y-intercepts. Let's convert Heather's equations into the slope-intercept form ($y = mx + b$), where $m$ is the slope and $b$ is the y-intercept.

For the first equation: $-6x + 18y = 0$ Add $6x$ to both sides: $18y = 6x$ Divide by $18$: $y = (6/18)x$ Simplify: $y = (1/3)x$ Here, the slope $m_1 = 1/3$ and the y-intercept $b_1 = 0$.

Now, for the second equation: $4x - 12y = 20$ Subtract $4x$ from both sides: $-12y = -4x + 20$ Divide by $-12$: $y = (-4/-12)x + (20/-12)$ Simplify: $y = (1/3)x - 5/3$ In this case, the slope $m_2 = 1/3$ and the y-intercept $b_2 = -5/3$.

See that? Both lines have the exact same slope of $1/3$! However, their y-intercepts are different ($0$ versus $-5/3$). This is the definitive proof that the lines are indeed parallel and distinct. They're heading in the same direction, but they start at different points and will never, ever cross. So, when Heather got $0=60$, it wasn't a mistake; it was the algebraic method brilliantly revealing this geometric fact. This outcome is super important because it distinguishes these systems from those with a single unique solution (intersecting lines) or infinitely many solutions (the same line). It's a fundamental concept in linear algebra that every student should grasp, as it provides a complete picture of the potential interactions between two linear equations.

What If You Get 0 = 0? (A Quick Detour)

Before we move on, it's worth taking a quick detour to discuss the other special case you might encounter when using the linear combination method: what if, instead of $0=60$, you ended up with $0=0$? This result, while also making both $x$ and $y$ variables vanish, carries a completely different meaning. When you get a mathematically true statement like $0=0$ (or $5=5$, or any number equaling itself), it signifies that the two equations are actually dependent and represent the exact same line. In simpler terms, the lines are coincident—they lie directly on top of each other. Think of it like drawing one line, and then drawing another line perfectly over it. How many points do they share? Well, every single point on the line! Because every point on the first line is also on the second line, such a system has infinitely many solutions. Any point $(x, y)$ that satisfies one equation will automatically satisfy the other. It means the equations are essentially different algebraic forms of the same underlying relationship. So, while $0=60$ screams "no solution!" and "parallel lines!", $0=0$ shouts "infinitely many solutions!" and "identical lines!" Knowing the difference between these two