Solve Systems Of Equations: Step-by-Step Guide

by Omar Yusuf 47 views

Hey guys! Today, we're diving into the fascinating world of solving systems of equations. This is a fundamental concept in mathematics, and mastering it opens doors to more advanced topics. We'll tackle a specific example, but the principles we discuss apply to a wide range of problems. So, buckle up and let's get started!

Understanding Systems of Equations

Before we jump into solving, let's first understand 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. The goal is to find values for these variables that satisfy all equations in the system simultaneously. Think of it like a puzzle where you need to find the perfect combination of numbers that makes all the pieces fit.

Each equation in the system represents a relationship between the variables. For example, in the given system:

2y - 3x = -27
5y + 3x = 6

the first equation tells us that twice the value of y minus three times the value of x equals -27. The second equation tells us that five times the value of y plus three times the value of x equals 6. We need to find values for x and y that make both of these statements true.

Systems of equations pop up everywhere in real-world applications. Imagine you're trying to figure out the cost of two different items when you know the total cost of a few combinations of those items. Or perhaps you're trying to determine the speeds of two cars traveling in different directions. Systems of equations provide the mathematical framework to solve these kinds of problems.

There are several methods for solving systems of equations, each with its own strengths and weaknesses. We'll focus on the elimination method in this example, but we'll also briefly touch on other methods like substitution. Understanding these different approaches allows you to choose the most efficient method for a given problem.

Now, why is it important to understand systems of equations? Well, they're not just abstract mathematical concepts. They are tools that help us model and solve real-world problems. From economics and engineering to computer science and physics, systems of equations are used to analyze relationships, make predictions, and optimize solutions. Mastering this skill gives you a powerful advantage in various fields.

Solving the System Using Elimination

Okay, let's dive into solving the specific system of equations we have:

2y - 3x = -27
5y + 3x = 6

The elimination method works by strategically adding or subtracting the equations in the system to eliminate one of the variables. Notice something interesting about our system: the coefficients of x in the two equations are -3 and +3. This is perfect! If we simply add the two equations together, the x terms will cancel out. This is the core idea behind elimination method.

Let's do it:

(2y - 3x) + (5y + 3x) = -27 + 6

Simplifying the left side, we get:

2y - 3x + 5y + 3x = -21

The -3x and +3x terms cancel each other out, leaving us with:

7y = -21

Now we have a simple equation with just one variable, y. To solve for y, we divide both sides of the equation by 7:

y = -21 / 7
y = -3

Great! We've found the value of y. But we're not done yet. We still need to find the value of x. This is where the second part of the elimination method comes in: back-substitution. Now that we know the value of y, we can substitute it into either of the original equations to solve for x. Let's choose the second equation, 5y + 3x = 6, but you can use either one – you'll get the same answer.

Substitute y = -3 into the equation:

5(-3) + 3x = 6
-15 + 3x = 6

Now we have an equation with only x. To solve for x, we first add 15 to both sides:

3x = 6 + 15
3x = 21

Then, we divide both sides by 3:

x = 21 / 3
x = 7

Fantastic! We've found the value of x as well. So, the solution to the system of equations is x = 7 and y = -3.

Verifying the Solution

It's always a good idea to verify your solution, especially in mathematics. It's like double-checking your work to make sure everything adds up. To verify our solution, we'll substitute the values of x and y back into both of the original equations. If the equations hold true, then we know our solution is correct.

Let's start with the first equation, 2y - 3x = -27. Substitute x = 7 and y = -3:

2(-3) - 3(7) = -27
-6 - 21 = -27
-27 = -27

The equation holds true! Now let's check the second equation, 5y + 3x = 6. Substitute x = 7 and y = -3:

5(-3) + 3(7) = 6
-15 + 21 = 6
6 = 6

This equation also holds true! Since our values for x and y satisfy both equations, we can confidently say that our solution is correct. It is very important to confirm the result obtained, to be sure that there was no math error in any of the previous steps.

Other Methods for Solving Systems of Equations

As I mentioned earlier, the elimination method isn't the only way to solve systems of equations. Another popular method is substitution. In the substitution method, you solve one equation for one variable in terms of the other variable, and then substitute that expression into the other equation. This reduces the system to a single equation with one variable, which you can then solve. Finally, you substitute the value you found back into one of the original equations to find the value of the other variable. The substitution method is particularly useful when one of the equations is already solved for one variable or can be easily solved for one variable.

There's also a graphical method for solving systems of equations. Each equation in the system represents a line on a graph. The solution to the system is the point where the lines intersect. This method is great for visualizing the solution, but it might not be as accurate as algebraic methods, especially if the solution involves fractions or decimals. The graphical method can be useful for understanding the concept of solutions to systems of equations, especially how multiple equations interact on a single plane.

The best method to use depends on the specific system of equations you're dealing with. For the system we solved today, elimination was a straightforward approach because the coefficients of x were opposites. But for other systems, substitution might be more efficient.

Why are Systems of Equations Important?

We've already touched on this a bit, but let's emphasize why systems of equations are so important. They're a powerful tool for modeling and solving real-world problems. Here are just a few examples:

  • Economics: Supply and demand curves can be represented as equations, and the equilibrium point (where supply equals demand) can be found by solving a system of equations.
  • Engineering: Circuit analysis often involves solving systems of equations to determine currents and voltages in different parts of the circuit.
  • Physics: Many physics problems, such as projectile motion and force analysis, require solving systems of equations.
  • Computer Science: Linear algebra, which heavily relies on systems of equations, is fundamental to many areas of computer science, including computer graphics, machine learning, and data analysis.

Understanding systems of equations opens up a world of possibilities. It's a foundational skill that you'll use in many different contexts, both in and out of the classroom.

Practice Makes Perfect

The best way to master solving systems of equations is to practice! Work through different examples, try different methods, and don't be afraid to make mistakes. Mistakes are learning opportunities. The more you practice, the more comfortable and confident you'll become.

So, there you have it! We've tackled a system of equations using the elimination method, verified our solution, and discussed other methods and the importance of this concept. Keep practicing, and you'll be a system-solving pro in no time!

Final Answer:

x = 7
y = -3