Solving Linear Systems Infinite Solutions For Y = -6x + 2 And -12x - 2y = -4
Hey guys! Let's dive into the exciting world of linear systems and figure out how many solutions a particular system has. We'll be looking at the system:
y = -6x + 2
-12x - 2y = -4
Our mission is to determine whether this system 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 this specific system, let's take a moment to understand what linear systems are and the different types of solutions they can have. A linear system is a set of two or more linear equations involving the same variables. The solution to a linear system is the set of values for the variables that make all the equations in the system true simultaneously. Graphically, a solution represents the point(s) where the lines intersect.
When we talk about solutions to linear systems, there are three possibilities:
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One Solution: This occurs when the lines intersect at a single point. This point represents the unique solution that satisfies both equations. Think of it as the lines crossing paths just once in the entire coordinate plane.
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No Solution: This happens when the lines are parallel and never intersect. In this case, there are no values for the variables that can satisfy both equations at the same time. Imagine two train tracks running perfectly parallel—they never meet, and neither do the solutions in this scenario.
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Infinite Solutions: This occurs when the equations represent the same line. In other words, one equation is a multiple of the other. Any point on the line will satisfy both equations, leading to an infinite number of solutions. Picture two lines drawn right on top of each other; they share every single point.
Understanding these possibilities is crucial for tackling our specific linear system. Now that we have a solid foundation, let's roll up our sleeves and get to work!
Analyzing the Given System: y = -6x + 2 and -12x - 2y = -4
Alright, let's get our hands dirty with the system we've got:
y = -6x + 2
-12x - 2y = -4
To figure out the number of solutions, we need to manipulate these equations and see how they relate to each other. There are a couple of ways we can approach this, but one common method is to try and get both equations into the same form. This will make it easier to compare them and identify any relationships.
Method 1: Substitution
Substitution is a powerful technique where we solve one equation for one variable and then substitute that expression into the other equation. This reduces the system to a single equation with a single variable, which is much easier to solve.
In our case, the first equation, y = -6x + 2, is already solved for y. This makes substitution a breeze. We can take this expression for y and plug it into the second equation:
-12x - 2(-6x + 2) = -4
Now, let's simplify this equation by distributing the -2:
-12x + 12x - 4 = -4
Notice anything interesting? The -12x and +12x cancel each other out, leaving us with:
-4 = -4
This is a true statement! But what does it mean in the context of our system? It means that the equation is always true, regardless of the values of x and y. This is a big clue that we might have infinitely many solutions.
Method 2: Elimination
Elimination is another fantastic method for solving linear systems. The idea here is to manipulate the equations so that the coefficients of one of the variables are opposites. When we add the equations together, that variable will be eliminated, leaving us with a single equation in one variable.
Let's rewrite our system:
y = -6x + 2
-12x - 2y = -4
To make the elimination method work smoothly, let's rearrange the first equation to match the form of the second equation. We can add 6x to both sides:
6x + y = 2
-12x - 2y = -4
Now, we want to make the coefficients of either x or y opposites. Let's focus on x. We can multiply the entire first equation by 2:
2(6x + y) = 2(2)
This gives us:
12x + 2y = 4
-12x - 2y = -4
Now, we have a beautiful setup for elimination. Notice that the coefficients of x are 12 and -12, and the coefficients of y are 2 and -2. When we add these equations together, both x and y will be eliminated:
(12x + 2y) + (-12x - 2y) = 4 + (-4)
This simplifies to:
0 = 0
Again, we've arrived at a true statement that doesn't involve any variables. This reinforces our suspicion that the system has infinitely many solutions.
Method 3: Transforming to Slope-Intercept Form
A third way to analyze the system is to convert both equations into slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. This form makes it easy to visually compare the lines represented by the equations.
The first equation, y = -6x + 2, is already in slope-intercept form. The slope is -6, and the y-intercept is 2.
Now, let's transform the second equation, -12x - 2y = -4, into slope-intercept form. First, we'll isolate the y term by adding 12x to both sides:
-2y = 12x - 4
Next, we'll divide both sides by -2:
y = -6x + 2
Guess what? We ended up with the exact same equation as the first one! This means that both equations represent the same line. When two equations represent the same line, they have infinitely many points in common, and therefore, the system has an infinite number of solutions.
Conclusion: Infinite Solutions
After analyzing the system using three different methods – substitution, elimination, and transforming to slope-intercept form – we've consistently arrived at the same conclusion: the linear system:
y = -6x + 2
-12x - 2y = -4
has an infinite number of solutions. This is because the two equations represent the same line. Any point on this line will satisfy both equations, leading to an endless set of solutions.
So, the correct answer is D. infinite number of solutions.
Isn't it fascinating how different approaches can lead us to the same answer? Linear systems are a fundamental concept in mathematics, and understanding how to solve them is a valuable skill. Keep practicing, guys, and you'll become masters of linear systems in no time!