Solving Systems Of Equations 2.5y + 3x = 27 And 5x - 2.5y = 5

Hey guys! Today, we're diving deep into solving systems of equations. Specifically, we're going to tackle this system:

$$\begin{array}{l} 2. 5 y+3 x=27 \\ 3. x-2.5 y=5 \end{array}$$

We'll break down how to solve it step-by-step and explore the result of adding the two equations together. So, buckle up, and let's get started!

Understanding Systems of Equations

Before we jump into solving, let's quickly recap what a system of equations 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 those variables that satisfy all equations in the system simultaneously. In simpler terms, we're looking for the point (or points) where the lines or curves represented by the equations intersect.

There are several methods we can use to solve systems of equations, including substitution, elimination, and graphing. In this case, we'll primarily focus on the elimination method, as it's particularly well-suited for this specific system.

Solving the System Using Elimination

The elimination method involves manipulating the equations in the system so that when we add them together, one of the variables cancels out. This leaves us with a single equation in one variable, which we can easily solve. Let's apply this to our system:

$$\begin{array}{l} 2. 5 y+3 x=27 \\ 3. x-2.5 y=5 \end{array}$$

Notice that the coefficients of the y terms are 2.5 and -2.5. This is perfect! If we add the equations together as they are, the y terms will conveniently cancel out. Let's do it:

(2.5y + 3x) + (5x - 2.5y) = 27 + 5

Combining like terms, we get:

8x = 32

Now, we can solve for x by dividing both sides by 8:

x = 32 / 8
x = 4

Great! We've found that x = 4. Now, we need to find the value of y. To do this, we can substitute the value of x back into either of the original equations. Let's use the first equation:

2.5y + 3(4) = 27

Simplify:

2.  5y + 12 = 27

Subtract 12 from both sides:

2.  5y = 15

Divide both sides by 2.5:

y = 15 / 2.5
y = 6

So, we've found that y = 6. Therefore, the solution to the system of equations is x = 4 and y = 6, which we can write as the ordered pair (4, 6).

Verification

It's always a good idea to verify our solution by plugging the values of x and y back into both original equations to make sure they hold true. Let's check:

Equation 1:

2.  5(6) + 3(4) = 15 + 12 = 27  (Correct!)

Equation 2:

5(4) - 2.5(6) = 20 - 15 = 5  (Correct!)

Since our solution satisfies both equations, we're confident that (4, 6) is indeed the correct solution.

The Result of Adding the Two Equations: A Closer Look

Now, let's address the second part of the question: What equation is the result of adding the two equations? We actually already did this when we solved the system! When we added the two equations together, we got:

(2.5y + 3x) + (5x - 2.5y) = 27 + 5

Which simplified to:

8x = 32

So, the equation that results from adding the two equations is 8x = 32. This equation is a linear equation in one variable (x), and it represents a vertical line on a graph. This equation is crucial because it allowed us to isolate x and find its value.

Why Does This Work? The Magic of Elimination

The elimination method works because we're essentially using the addition property of equality. This property states that if we add the same quantity to both sides of an equation, the equation remains balanced. In our case, we're adding the entire left-hand side of one equation to the left-hand side of the other, and the entire right-hand side of one equation to the right-hand side of the other. This maintains the equality.

The key to the elimination method is to strategically manipulate the equations so that when we add them, one of the variables cancels out. This simplifies the problem and allows us to solve for the remaining variable.

Alternative Approaches: Substitution Method

While we used the elimination method here, it's worth mentioning that we could also solve this system using the substitution method. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation.

For example, we could solve the second equation for x:

5x - 2.5y = 5
5x = 2.5y + 5
x = (2.5y + 5) / 5
x = 0.5y + 1

Then, we could substitute this expression for x into the first equation:

2.  5y + 3(0.5y + 1) = 27

And solve for y. Once we find y, we can substitute it back into the expression for x to find the value of x. While the substitution method works, in this case, the elimination method was arguably more straightforward because the y terms were already set up to cancel out.

Real-World Applications of Systems of Equations

Systems of equations aren't just abstract mathematical concepts; they have numerous real-world applications. They're used in fields like:

• Physics: Solving for forces, velocities, and accelerations.
• Engineering: Designing structures and circuits.
• Economics: Modeling supply and demand.
• Computer Science: Developing algorithms and simulations.
• Chemistry: Balancing chemical equations.

For example, consider a scenario where you're trying to determine the cost of two different items. You know the total cost of a combination of the items and the total cost of a different combination. You can set up a system of equations to represent this situation and solve for the individual costs of the items. Think about mixing solutions in chemistry, designing electrical circuits, or even planning a budget! These are just a few scenarios where understanding and solving systems of equations can come in handy.

Common Mistakes to Avoid

When solving systems of equations, it's easy to make mistakes if you're not careful. Here are a few common pitfalls to watch out for:

1. Arithmetic Errors: Be meticulous with your calculations! A simple arithmetic mistake can throw off your entire solution.
2. Incorrectly Distributing: When multiplying an equation by a constant, make sure you distribute the constant to all terms in the equation.
3. Forgetting to Substitute Back: After solving for one variable, remember to substitute that value back into one of the original equations to solve for the other variable.
4. Not Checking Your Solution: Always verify your solution by plugging the values back into both original equations.
5. Misinterpreting the Solution: Make sure you understand what the solution represents in the context of the problem. For example, if you're solving a word problem, make sure your answer makes sense.

By being aware of these common mistakes and taking your time, you can increase your accuracy and confidence in solving systems of equations.

Conclusion: Mastering Systems of Equations

So, there you have it! We've successfully solved the system of equations:

$$\begin{array}{l} 2. 5 y+3 x=27 \\ 3. x-2.5 y=5 \end{array}$$

And we found that the solution is x = 4 and y = 6, or (4, 6). We also determined that the equation resulting from adding the two equations is 8x = 32. Remember, the key to mastering systems of equations is practice, practice, practice! The more you work through different problems, the more comfortable and confident you'll become.

Understanding how to solve systems of equations is a valuable skill that can be applied in various fields. Whether you're a student tackling algebra problems or a professional working in science, engineering, or economics, the ability to solve systems of equations will serve you well. So, keep practicing, and don't be afraid to ask for help when you need it. You've got this!

If you guys have any questions or want to explore other methods, feel free to ask! Happy solving!