Solving System Of Equations -3x + 6y = -3 And 5x - 10y = 5 With Infinite Solutions

Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of linear equations, specifically focusing on a system of equations. We'll be tackling the following system:

-3x + 6y = -3
5x - 10y = 5

This looks like a classic case of two equations with two unknowns (x and y), and we're here to figure out the values of x and y that satisfy both equations simultaneously. So, buckle up, grab your thinking caps, and let's embark on this mathematical journey together!

Understanding the Basics of Systems of Equations

Before we jump into solving this particular system, let's quickly recap what a system of equations actually is. Think of it as a puzzle where you have multiple pieces of information (the equations) and you need to fit them together to find the solution (the values of the variables). In our case, we have two linear equations. A linear equation is simply an equation where the highest power of the variables is 1. When you graph these equations, they form straight lines – hence the name “linear.”

A system of equations is a set of two or more equations that share the same variables. The goal is to find the values for those variables that make all the equations in the system true at the same time. There are several methods we can use to solve systems of equations, including:

• Substitution: Solving one equation for one variable and then substituting that expression into the other equation.
• Elimination: Multiplying one or both equations by constants to make the coefficients of one variable opposites, and then adding the equations together to eliminate that variable.
• Graphing: Graphing both equations and finding the point where the lines intersect (the point of intersection represents the solution).

For this particular system, we'll explore the elimination method as it seems to be a very efficient approach. Let’s dive into the detailed steps.

Solving the System Using the Elimination Method

The elimination method is a powerful technique for solving systems of equations. The main idea is to manipulate the equations so that when you add them together, one of the variables cancels out, leaving you with a single equation in one variable. Here’s how we can apply it to our system:

-3x + 6y = -3
5x - 10y = 5

1. Identify a Variable to Eliminate: Look at the coefficients of x and y in both equations. Notice that if we multiply the first equation by 5 and the second equation by 3, the coefficients of x will become -15 and 15, respectively. This is perfect because when we add the equations, the x terms will cancel out.

2. Multiply the Equations:

   • Multiply the first equation (-3x + 6y = -3) by 5: This gives us -15x + 30y = -15.
   • Multiply the second equation (5x - 10y = 5) by 3: This gives us 15x - 30y = 15.

Now our system looks like this:

-15x + 30y = -15
15x - 30y = 15

3. Add the Equations: Add the two modified equations together, term by term:

   (-15x + 30y) + (15x - 30y) = -15 + 15
   

   Simplifying, we get:

   0 = 0
   

Wait a minute! We ended up with 0 = 0. What does this mean? This is a very important and interesting result, so let's interpret it in the next section.

Interpreting the Result: 0 = 0 – Infinite Solutions

Okay, so we performed the elimination method and ended up with the equation 0 = 0. This might seem a bit strange at first, but it actually tells us something very important about our system of equations. When you arrive at an identity (a statement that is always true, like 0 = 0 or 5 = 5), it means that the two equations in your system are essentially the same equation, just written in different forms.

Graphically, this means that the two lines represented by the equations are actually the same line. They completely overlap. Therefore, any point that lies on this line is a solution to both equations. Since there are infinitely many points on a line, this system has infinitely many solutions. This type of system is called a dependent system.

Think of it this way: imagine you're trying to find the intersection point of two lines, but the lines are right on top of each other. Every single point on the line is an intersection! This is exactly what’s happening in our case. This outcome highlights a critical point in solving systems of equations. Not all systems have a unique solution; some have no solutions, and some have infinitely many. Our given system falls into the latter category. Let's delve deeper to understand why these equations represent the same line.

Demonstrating the Equivalence of the Equations

To truly understand why we have infinitely many solutions, let’s show explicitly that the two equations in our system are equivalent. Remember our original system:

-3x + 6y = -3
5x - 10y = 5

Let's take the first equation, -3x + 6y = -3, and try to manipulate it to look like the second equation. We can start by dividing both sides of the first equation by -3:

(-3x + 6y) / -3 = -3 / -3
x - 2y = 1

Now, let's take the second equation, 5x - 10y = 5, and divide both sides by 5:

(5x - 10y) / 5 = 5 / 5
x - 2y = 1

Aha! We see that both equations simplify to the exact same equation: x - 2y = 1. This clearly demonstrates that the two original equations are just multiples of each other. They represent the same line, and that's why we obtained 0 = 0 when using the elimination method. This confirms that we have infinitely many solutions. To express these solutions, we can write them in parametric form, giving a general representation for all the points on the line.

Expressing the Infinite Solutions in Parametric Form

Since we have infinitely many solutions, we need a way to represent them. We can do this using parametric form. This involves expressing one variable in terms of a parameter (usually denoted by 't') and then expressing the other variable in terms of the same parameter. Let's use the simplified equation we found earlier, x - 2y = 1.

1. Solve for One Variable: Let's solve for x:

   x = 2y + 1
   

2. Introduce the Parameter: Let's let y = t, where t is any real number. This means we're treating y as our parameter.

3. Express the Other Variable in Terms of the Parameter: Substitute y = t into the equation we found for x:

   x = 2t + 1
   

So, the parametric form of the solutions is:

x = 2t + 1
y = t

This tells us that for any real number t, the point (2t + 1, t) is a solution to our system of equations. For example:

• If t = 0, we get the solution (1, 0).
• If t = 1, we get the solution (3, 1).
• If t = -1, we get the solution (-1, -1).

And so on! We can generate infinitely many solutions by simply plugging in different values for t. This parametric form beautifully captures the essence of having infinitely many solutions – a line of solutions, to be precise. Understanding parametric representation is crucial for dealing with such systems.

Graphical Representation and Verification

To further solidify our understanding, let's visualize this system graphically. If we were to graph the two equations, -3x + 6y = -3 and 5x - 10y = 5, we would see that they overlap perfectly, forming a single line. This visual confirmation supports our algebraic finding of infinitely many solutions.

You can use graphing software or an online graphing calculator to plot these equations. You'll notice that both equations produce the same line, affirming our conclusion that every point on the line is a solution to the system. This graphical representation acts as a powerful verification tool, strengthening our confidence in the mathematical results we’ve obtained through algebraic manipulation.

Moreover, plotting a few points generated from our parametric solution (like those we calculated earlier) onto the graph will also show that these points indeed lie on the line, providing yet another layer of validation.

Conclusion: The Beauty of Infinite Solutions

In conclusion, we've successfully unraveled the system of equations -3x + 6y = -3 and 5x - 10y = 5. By employing the elimination method, we discovered that this system has infinitely many solutions. This occurred because the two equations are equivalent, representing the same line. We then expressed these solutions in parametric form as x = 2t + 1 and y = t, providing a way to generate all possible solutions.

This journey through solving a system with infinite solutions highlights the rich diversity of possibilities in linear algebra. It's a reminder that not every system has a single, unique answer, and understanding how to interpret and represent these alternative outcomes is just as important. So, keep exploring, keep questioning, and keep embracing the fascinating world of mathematics, guys! We've seen how mathematical tools can elegantly describe a situation where solutions aren't just limited but infinite, showcasing the power and beauty inherent in mathematical concepts.