Solving 3x - 3y = -2 And -2x + 2y = 2 A Detailed Guide

Hey guys! Today, we're diving deep into the world of algebra to tackle a system of equations. Specifically, we'll be looking at how to solve the system:

• 3x - 3y = -2
• -2x + 2y = 2

This might look a bit intimidating at first, but don't worry! We're going to break it down step-by-step, so you'll be a pro at solving these in no time. We will explore different methods and provide detailed explanations so that you can understand not just the how, but also the why behind each step. 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 actually is. A system of equations is simply a set of two or more equations that contain the same variables. The goal is to find values for those 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 fit all the pieces.

In our case, we have two equations and two variables (x and y). There are several methods we can use to solve such systems, including substitution, elimination, and graphing. We'll primarily focus on the substitution and elimination methods in this guide. These methods allow us to manipulate the equations in ways that help isolate one variable, making it easier to solve for the other. It's like having different tools in a toolbox – each one is suited for different situations, and knowing when to use which tool is key to becoming a master equation solver.

The solutions to a system of equations can be one of three types:

1. Unique Solution: There is exactly one pair of (x, y) values that satisfy both equations. This means the lines represented by the equations intersect at a single point.
2. No Solution: The equations are inconsistent, meaning there are no values of x and y that can satisfy both equations simultaneously. This often happens when the lines represented by the equations are parallel and never intersect.
3. Infinitely Many Solutions: The equations are dependent, meaning they represent the same line. Any (x, y) pair that satisfies one equation will also satisfy the other. This is like having two identical puzzle pieces – they fit together perfectly, but don't give you any new information.

Method 1: Elimination Method

The elimination method, also known as the addition method, involves manipulating the equations so that when you add them together, one of the variables cancels out. This leaves you with a single equation in one variable, which you can easily solve. To apply this method effectively, you often need to multiply one or both equations by a constant to make the coefficients of one variable opposites.

Let's apply the elimination method to our system:

1. 3x - 3y = -2
2. -2x + 2y = 2

Notice that the coefficients of y are -3 and 2. To eliminate y, we need to find a common multiple of 3 and 2, which is 6. We can multiply the first equation by 2 and the second equation by 3 to make the coefficients of y equal to -6 and 6, respectively. This way, when we add the equations, the y terms will cancel out.

Multiply equation (1) by 2:

2 * (3x - 3y) = 2 * (-2)

6x - 6y = -4

Multiply equation (2) by 3:

3 * (-2x + 2y) = 3 * (2)

-6x + 6y = 6

Now we have the modified system:

1. 6x - 6y = -4
2. -6x + 6y = 6

Next, add the two equations together:

(6x - 6y) + (-6x + 6y) = -4 + 6

6x - 6x - 6y + 6y = 2

0 = 2

Wait a minute! We ended up with the equation 0 = 2, which is clearly not true. This tells us something important: the system of equations has no solution. The lines represented by these equations are parallel and never intersect.

This outcome is a critical part of understanding systems of equations. Sometimes, the math leads us to a contradiction, and that contradiction is the key to the solution. It's like a detective story where the inconsistency is the clue that unravels the mystery.

Method 2: Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This effectively reduces the system to a single equation in one variable. This method is particularly useful when one of the equations can be easily solved for one variable in terms of the other. For instance, if you have an equation like y = 2x + 3, substituting that expression for y in the other equation can simplify things greatly.

Let's try using the substitution method on our system:

1. 3x - 3y = -2
2. -2x + 2y = 2

First, let's simplify equation (2) by dividing both sides by 2:

(-2x + 2y) / 2 = 2 / 2

-x + y = 1

Now, solve this simplified equation for y:

y = x + 1

Next, substitute this expression for y into equation (1):

3x - 3(x + 1) = -2

Now, distribute the -3:

3x - 3x - 3 = -2

Combine like terms:

-3 = -2

Again, we arrive at a contradiction! The equation -3 = -2 is not true. This confirms our previous finding that the system of equations has no solution. The substitution method provided an alternative path to the same conclusion, reinforcing the fact that the lines are parallel.

Graphical Interpretation

To further understand why this system has no solution, let's consider the graphical interpretation. Each equation in the system represents a line in the coordinate plane. The solution to the system is the point (or points) where the lines intersect. If the lines are parallel, they never intersect, and the system has no solution.

Let's rewrite our equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

Equation (1): 3x - 3y = -2

Subtract 3x from both sides:

-3y = -3x - 2

Divide by -3:

y = x + 2/3

Equation (2): -2x + 2y = 2

Add 2x to both sides:

2y = 2x + 2

Divide by 2:

y = x + 1

Notice that both equations have the same slope (m = 1), but different y-intercepts (2/3 and 1). This means the lines are parallel and will never intersect. Graphing these lines would visually confirm that they run side-by-side without ever meeting, solidifying our understanding of why there's no solution to this system.

Checking for Consistency and Dependency

When solving systems of equations, it's crucial to check for consistency and dependency. This involves analyzing the equations to determine whether they have a solution (consistent) or not (inconsistent), and whether they are independent or dependent.

• Consistent System: A system is consistent if it has at least one solution (either a unique solution or infinitely many solutions).
• Inconsistent System: A system is inconsistent if it has no solution.
• Independent Equations: Equations are independent if they represent distinct lines.
• Dependent Equations: Equations are dependent if they represent the same line.

In our case, the system is inconsistent because we found no solution. The equations are also independent because they represent different lines (even though they are parallel).

Understanding these concepts helps in predicting the nature of the solutions before diving into the actual solving process. It’s like having a roadmap before embarking on a journey – it gives you a sense of direction and helps you anticipate potential roadblocks.

Common Mistakes to Avoid

When solving systems of equations, it's easy to make small mistakes that can lead to incorrect answers. Here are some common pitfalls to watch out for:

1. Incorrectly Distributing: Make sure to distribute numbers correctly when multiplying equations. For example, if you have 2(x - y), it should be 2x - 2y, not 2x - y.
2. Sign Errors: Pay close attention to signs, especially when adding or subtracting equations. A simple sign mistake can throw off the entire solution.
3. Arithmetic Errors: Double-check your arithmetic calculations, especially when dealing with fractions or negative numbers. A small arithmetic error can propagate through the rest of the problem.
4. Not Checking Your Solution: Always plug your solution back into the original equations to verify that it satisfies both equations. This is a crucial step in catching errors.
5. Misinterpreting No Solution: When you arrive at a contradiction (like 0 = 2), it means the system has no solution, not that you made a mistake. This is a valid outcome.

By being mindful of these common mistakes, you can significantly improve your accuracy and confidence in solving systems of equations. It’s like learning the rules of the road – knowing the common pitfalls helps you navigate safely and efficiently.

Conclusion

So, to wrap it up, we've shown that the system of equations:

• 3x - 3y = -2
• -2x + 2y = 2

has no solution. We arrived at this conclusion using both the elimination and substitution methods, and we confirmed it by analyzing the slopes and y-intercepts of the lines. Remember, when you encounter a contradiction while solving, it's a sign that the system is inconsistent.

Solving systems of equations is a fundamental skill in algebra and has applications in many areas of mathematics and real-world problems. By mastering these methods and understanding the concepts of consistency and dependency, you'll be well-equipped to tackle more complex problems in the future. Keep practicing, guys, and you'll become equation-solving pros in no time!