Solving 3x - 2y = 9 And 3x + 2y = -12 A Comprehensive Guide
Solving systems of equations is a fundamental concept in mathematics, and it's something you'll encounter frequently in various fields. Today, we're going to break down a specific problem: solving the system of equations 3x - 2y = 9 and 3x + 2y = -12. We'll explore different methods, understand the underlying principles, and see how this knowledge can be applied in real-world scenarios. So, let's dive in and unravel this mathematical puzzle together, guys!
Understanding the Problem
Before we jump into solving, let's make sure we understand what we're dealing with. We have two equations:
- 3x - 2y = 9
- 3x + 2y = -12
Each of these equations represents a straight line when graphed on a coordinate plane. The solution to this system of equations is the point (x, y) where these two lines intersect. In other words, it's the pair of values for x and y that satisfy both equations simultaneously. Think of it like finding the common ground between these two mathematical statements.
There are several methods we can use to find this solution, and we'll explore the most common ones: the elimination method and the substitution method. Understanding these methods is crucial, not just for solving this specific problem, but for tackling a wide range of mathematical challenges. It's like having different tools in your toolbox – each one is useful in different situations. So, let's get our hands dirty and see how these methods work!
Method 1: The Elimination Method
The elimination method, also known as the addition method, is a powerful technique for solving systems of equations. The core idea behind this method is to manipulate the equations in such a way that when we add them together, one of the variables cancels out, leaving us with a single equation in a single variable. This simplifies the problem significantly, making it easier to solve. It's like magic, but it's actually math!
In our case, notice something interesting about the equations:
- 3x - 2y = 9
- 3x + 2y = -12
The coefficients of the 'y' terms are opposites: -2 and +2. This is exactly what we need for the elimination method to work smoothly! When we add these two equations together, the 'y' terms will cancel out, leaving us with an equation in just 'x'.
Let's perform the addition:
(3x - 2y) + (3x + 2y) = 9 + (-12)
Simplifying the equation, we get:
6x = -3
Now, we have a simple equation to solve for 'x'. Divide both sides by 6:
x = -3/6
x = -1/2
So, we've found the value of 'x'! Awesome, right? Now, we need to find the value of 'y'. We can do this by substituting the value of 'x' back into either of the original equations. It doesn't matter which equation we choose; we'll get the same answer for 'y' either way.
Let's substitute x = -1/2 into the first equation, 3x - 2y = 9:
3(-1/2) - 2y = 9
Simplifying, we get:
-3/2 - 2y = 9
To solve for 'y', we first add 3/2 to both sides:
-2y = 9 + 3/2
-2y = 18/2 + 3/2
-2y = 21/2
Now, divide both sides by -2:
y = (21/2) / (-2)
y = -21/4
Therefore, the solution to the system of equations using the elimination method is x = -1/2 and y = -21/4. This means the point of intersection of the two lines represented by these equations is (-1/2, -21/4). We nailed it!
Method 2: The Substitution Method
The substitution method is another powerful technique for solving systems of equations. In this method, we solve one equation for one variable in terms of the other variable, and then substitute that expression into the second equation. This again leaves us with a single equation in a single variable, which we can then solve. Think of it as replacing one piece of the puzzle with an equivalent piece.
Let's revisit our system of equations:
- 3x - 2y = 9
- 3x + 2y = -12
In this case, neither equation is immediately easier to solve for a single variable. However, let's choose the first equation (3x - 2y = 9) and solve it for 'x'. This choice is somewhat arbitrary; we could have chosen the second equation or solved for 'y' instead. The beauty of math is that there are often multiple paths to the same destination!
To solve for 'x', we first add 2y to both sides:
3x = 9 + 2y
Now, divide both sides by 3:
x = (9 + 2y) / 3
So, we have expressed 'x' in terms of 'y'. Now comes the substitution part. We substitute this expression for 'x' into the second equation (3x + 2y = -12):
3((9 + 2y) / 3) + 2y = -12
Notice that the '3' in the numerator and denominator cancels out:
9 + 2y + 2y = -12
Simplifying, we get:
9 + 4y = -12
Now, subtract 9 from both sides:
4y = -21
Finally, divide both sides by 4:
y = -21/4
Hey, that looks familiar! We got the same value for 'y' as we did with the elimination method. This is a good sign – it confirms that our solution is consistent.
Now that we have the value of 'y', we can substitute it back into the expression we found for 'x':
x = (9 + 2y) / 3
x = (9 + 2(-21/4)) / 3
Simplifying:
x = (9 - 21/2) / 3
x = (18/2 - 21/2) / 3
x = (-3/2) / 3
x = -1/2
And there it is! We've found the same solution for 'x' as before. The solution to the system of equations using the substitution method is x = -1/2 and y = -21/4, which matches the result we obtained using the elimination method. This reinforces our confidence in the solution and demonstrates the flexibility of different problem-solving approaches.
Verifying the Solution
It's always a good practice to verify our solution, regardless of the method we used. This helps us catch any potential errors and ensures that our answer is correct. To verify, we simply substitute the values of x and y back into both original equations and see if they hold true.
Let's substitute x = -1/2 and y = -21/4 into the first equation, 3x - 2y = 9:
3(-1/2) - 2(-21/4) = 9
Simplifying:
-3/2 + 21/2 = 9
18/2 = 9
9 = 9
The first equation holds true! That's a good start.
Now, let's substitute the values into the second equation, 3x + 2y = -12:
3(-1/2) + 2(-21/4) = -12
Simplifying:
-3/2 - 21/2 = -12
-24/2 = -12
-12 = -12
Excellent! The second equation also holds true. Since the values of x and y satisfy both equations, we can confidently say that our solution is correct.
Graphical Interpretation
As we mentioned earlier, each equation in a system of equations represents a line on a coordinate plane. The solution to the system is the point where these lines intersect. Let's visualize this to get a deeper understanding.
If we were to graph the lines represented by the equations 3x - 2y = 9 and 3x + 2y = -12, we would find that they intersect at the point (-1/2, -21/4). This point is the graphical representation of our solution.
Graphing systems of equations can be a helpful way to visualize the solutions and to understand the relationship between the equations. It also provides a visual check for our algebraic solutions. Sometimes, seeing is believing!
Real-World Applications
Solving systems of equations isn't just an abstract mathematical exercise; it has numerous applications in the real world. These systems can be used to model and solve problems in various fields, including:
- Engineering: Calculating forces, stresses, and strains in structures.
- Economics: Determining equilibrium prices and quantities in markets.
- Physics: Analyzing motion and interactions of objects.
- Computer Science: Solving optimization problems and designing algorithms.
- Chemistry: Balancing chemical equations.
For example, imagine you're running a business and you need to determine the optimal pricing for two products to maximize your profit. You can set up a system of equations to model the relationship between price, demand, and profit, and then solve the system to find the optimal prices. Pretty cool, huh?
The ability to solve systems of equations is a valuable skill that can be applied in a wide range of situations. It's a cornerstone of mathematical problem-solving and a key tool for anyone working in a quantitative field. So, keep practicing and honing your skills!
Conclusion
In this article, we've taken a comprehensive look at solving the system of equations 3x - 2y = 9 and 3x + 2y = -12. We explored two common methods: the elimination method and the substitution method. We found that both methods lead to the same solution: x = -1/2 and y = -21/4. We also verified our solution and discussed the graphical interpretation and real-world applications of solving systems of equations.
Understanding these concepts and techniques is essential for anyone studying mathematics or working in a related field. The ability to solve systems of equations is a powerful tool that can be used to tackle a wide variety of problems. So, keep exploring, keep learning, and keep solving! You've got this!
