Solving Systems Of Equations: A Step-by-Step Guide For X + 3y = 9 And 3x - 3y = -13

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Hey guys! Today, we're diving deep into the world of algebra to tackle a classic problem: solving a system of linear equations. Specifically, we're going to break down the equations $x + 3y = 9$ and $3x - 3y = -13$. Whether you're a student brushing up on your skills or just someone who enjoys a good math puzzle, this guide will walk you through a clear, step-by-step solution. So, grab your pencils, and let's get started!

Understanding the Basics of Systems of Equations

Before we jump into solving, let's quickly recap what a system of equations actually is. A system of equations is simply a set of two or more equations that share the same variables. Our goal is to find the values of these variables that satisfy all equations in the system simultaneously. Think of it like finding the perfect puzzle pieces that fit together in multiple puzzles at once. There are several methods to solve these systems, including substitution, elimination, and graphing. Each method has its strengths, and the best one to use often depends on the specific equations you're dealing with. For the equations at hand, we'll be focusing on the elimination method, which will become clear shortly.

Systems of equations are a fundamental concept in algebra and have a wide range of applications in real-world scenarios. From calculating the break-even point in business to modeling physical phenomena in science and engineering, understanding how to solve these systems is incredibly valuable. These systems often represent real-world constraints and relationships. For instance, in economics, you might use a system of equations to determine the equilibrium price and quantity in a market where supply and demand intersect. In physics, you might use them to describe the motion of multiple objects or the forces acting on a system. The beauty of algebra lies in its ability to translate these complex situations into mathematical models that we can then solve using techniques like the ones we'll explore today. This ability to model and solve real-world problems is what makes the study of systems of equations so powerful and relevant.

When solving systems of equations, we are essentially looking for the point (or points) where the lines represented by the equations intersect. Each linear equation in the system represents a straight line on a graph. The solution to the system is the set of coordinates (x, y) that lie on both lines. There are three possible outcomes when solving a system of two linear equations: one unique solution (the lines intersect at one point), no solution (the lines are parallel and never intersect), or infinitely many solutions (the lines are the same, overlapping each other). Recognizing which outcome is possible based on the equations themselves is a crucial skill. For instance, if the equations have the same slope but different y-intercepts, you know the lines are parallel and there will be no solution. If the equations are multiples of each other, you know they represent the same line and there will be infinitely many solutions. In our case, we'll see that the equations $x + 3y = 9$ and $3x - 3y = -13$ intersect at a single point, giving us a unique solution. This understanding of the graphical representation helps to solidify the concept and provides a visual way to interpret the algebraic steps we take to find the solution.

The Elimination Method: A Step-by-Step Approach

For our system of equations, the elimination method is a particularly efficient way to find the solution. This method involves manipulating the equations so that when we add them together, one of the variables cancels out, leaving us with a single equation in one variable. This makes it much easier to solve. Let's dive into how it works with our equations:

$x + 3y = 9$ (Equation 1)

$3x - 3y = -13$ (Equation 2)

Notice anything interesting? The coefficients of the $y$ terms are $3$ and $-3$. This is perfect because if we add these equations together, the $y$ terms will eliminate each other!

The elimination method shines when you can easily identify terms that will cancel out when equations are added or subtracted. The key is to strategically manipulate the equations, usually by multiplying one or both equations by a constant, so that the coefficients of one variable are opposites. In our case, we were lucky – the $y$ terms already had opposite coefficients. But often, you'll need to do some preliminary work. For example, if you had equations like $2x + y = 5$ and $x - 3y = 2$, you might multiply the second equation by -2 to eliminate the $x$ terms, or multiply the first equation by 3 to eliminate the $y$ terms. The choice depends on which variable you find easier to eliminate. Once you've manipulated the equations, you can add or subtract them to eliminate a variable, leaving you with a single equation in one variable. This is the core idea behind the elimination method, and it's a powerful technique for solving systems of equations. It’s also worth noting that the elimination method is closely related to the concept of linear combinations in linear algebra, providing a deeper theoretical foundation for this technique.

To further illustrate the power of the elimination method, consider a scenario where you have more complex coefficients. Suppose you have the equations $4x + 5y = 12$ and $3x - 2y = 5$. In this case, you'd need to multiply both equations to eliminate a variable. For instance, to eliminate $x$, you could multiply the first equation by 3 and the second equation by -4, resulting in $12x + 15y = 36$ and $-12x + 8y = -20$. Adding these equations would then eliminate $x$, leaving you with $23y = 16$, which you can solve for $y$. The beauty of this approach is that it can handle any system of linear equations, regardless of how complicated the coefficients are. Moreover, the elimination method is not just limited to systems of two equations. It can be extended to systems with three or more equations and variables. In such cases, you systematically eliminate variables one by one until you are left with a single equation in one variable. This makes the elimination method a versatile and robust tool for solving a wide range of linear systems.

Step 1: Adding the Equations

Let's add Equation 1 and Equation 2:

$(x + 3y) + (3x - 3y) = 9 + (-13)$

Simplifying this, we get:

$4x = -4$

Step 2: Solving for $x$

Now, we have a simple equation with just one variable. To solve for $x$, divide both sides by 4:

$x = -1$

Great! We've found the value of $x$. Now, let's use this value to find $y$.

Step 3: Substituting $x$ into Equation 1

We can substitute $x = -1$ into either Equation 1 or Equation 2. Let's use Equation 1:

$(-1) + 3y = 9$

Add 1 to both sides:

$3y = 10$

Step 4: Solving for $y$

Divide both sides by 3:

y = rac{10}{3}

So, we've found that y = rac{10}{3}.

Step 5: The Solution

Our solution to the system of equations is $x = -1$ and y = rac{10}{3}. We can write this as an ordered pair: (-1, rac{10}{3}).

To ensure accuracy in solving systems of equations, it's always a good idea to check your solution by substituting the values back into the original equations. This step helps catch any potential errors in your calculations and confirms that the solution indeed satisfies both equations. Let's substitute $x = -1$ and y = rac{10}{3} into our original equations:

For Equation 1: $x + 3y = 9$

(-1) + 3(rac{10}{3}) = -1 + 10 = 9

This checks out!

For Equation 2: $3x - 3y = -13$

3(-1) - 3(rac{10}{3}) = -3 - 10 = -13

This also checks out! Since our solution satisfies both equations, we can confidently say that $x = -1$ and y = rac{10}{3} is the correct solution to the system. This verification step is particularly important when dealing with more complex systems or when you've performed multiple operations to solve the equations. It provides a safeguard against errors and ensures that your final answer is accurate. Furthermore, the act of checking your solution reinforces your understanding of the problem and the steps you took to solve it, contributing to a deeper comprehension of the underlying concepts.

Checking Our Solution

It's always a good idea to check our work. Let's plug our values back into the original equations:

For Equation 1: -1 + 3(rac{10}{3}) = -1 + 10 = 9 (Correct!)

For Equation 2: 3(-1) - 3(rac{10}{3}) = -3 - 10 = -13 (Correct!)

Our solution checks out. Awesome!

Alternative Methods: Substitution

While we used elimination, it's worth mentioning the substitution method as another approach to solving systems of equations. In substitution, you solve one equation for one variable and then substitute that expression into the other equation. This leaves you with a single equation in one variable, which you can solve. Then, you substitute the value you found back into one of the original equations to find the other variable. Let’s briefly illustrate how this could work with our equations.

From Equation 1 ($x + 3y = 9$), we can solve for $x$:

$x = 9 - 3y$

Now, substitute this expression for $x$ into Equation 2 ($3x - 3y = -13$):

$3(9 - 3y) - 3y = -13$

Simplify and solve for $y$:

$27 - 9y - 3y = -13$

$27 - 12y = -13$

$-12y = -40$

y = rac{10}{3}

Now, substitute y = rac{10}{3} back into the expression for $x$:

x = 9 - 3(rac{10}{3})

$x = 9 - 10$

$x = -1$

As you can see, we arrive at the same solution ($x = -1$, y = rac{10}{3}) using the substitution method. The choice between elimination and substitution often comes down to personal preference or the specific structure of the equations. Some systems are more naturally suited to one method over the other. For example, if one equation is already solved for one variable, substitution might be the easier route. However, the elimination method can be particularly efficient when coefficients of one variable are easily made opposites, as was the case in our example. Understanding both methods gives you flexibility and a broader toolkit for tackling different types of systems of equations.

Real-World Applications

Systems of equations aren't just abstract math problems; they pop up in numerous real-world situations. Think about scenarios where you have multiple constraints or relationships between variables. For example:

• Economics: Determining market equilibrium, where supply equals demand.
• Physics: Calculating the motion of objects under multiple forces.
• Chemistry: Balancing chemical equations.
• Engineering: Designing structures with specific load-bearing requirements.
• Finance: Portfolio optimization and investment strategies.

Let's consider a simple example. Suppose you're planning a party and have a budget of $200. You want to buy pizzas that cost $15 each and drinks that cost $2 each. You also want to have a total of 20 items (pizzas and drinks) for your guests. You can set up a system of equations to determine how many pizzas and drinks you should buy.

Let $p$ be the number of pizzas and $d$ be the number of drinks. The budget constraint can be represented as:

$15p + 2d = 200$ (Equation 1)

The total items constraint can be represented as:

$p + d = 20$ (Equation 2)

You can solve this system of equations using either elimination or substitution to find the optimal number of pizzas and drinks to buy while staying within your budget and providing enough items for your guests. This simple example illustrates how systems of equations can help you make informed decisions in everyday situations. The ability to translate real-world problems into mathematical models and solve them is a powerful skill that has wide-ranging applications in various fields.

Conclusion

We've successfully solved the system of equations $x + 3y = 9$ and $3x - 3y = -13$ using the elimination method. Our solution is $x = -1$ and y = rac{10}{3}. Remember, practice makes perfect, so keep solving those equations! Understanding systems of equations is a crucial step in your mathematical journey, opening doors to more advanced concepts and real-world problem-solving. Keep up the great work, and you'll be an algebra ace in no time! And remember, math can be fun – it's all about finding the right approach and breaking down the problem into manageable steps. You've got this!