Solve 2x + 3y = 12 And X - Y = 1: Find The Solution
Hey there, math enthusiasts! Today, we're diving into a classic problem in algebra: solving a system of equations. Specifically, we'll be tackling the system:
2x + 3y = 12
x - y = 1
Our mission? To find the ordered pair (x, y) that satisfies both equations simultaneously. This is a fundamental concept in mathematics, and mastering it will unlock doors to more advanced topics. So, grab your pencils, and let's get started!
Understanding Systems of Equations
Before we jump into the solution, let's take a moment to understand what a system of equations really means. A system of equations is simply a set of two or more equations that contain the same variables. In our case, we have two equations with two variables, x and y. The solution to a system of equations is the set of values for the variables that make all the equations true at the same time. Geometrically, each equation represents a line on a graph, and the solution to the system is the point where the lines intersect.
Solving systems of equations is crucial in many real-world applications, guys. Think about situations where you have multiple constraints or conditions, and you need to find values that satisfy all of them. For example, businesses use systems of equations to optimize production costs and maximize profits. Scientists use them to model complex phenomena, and engineers use them to design structures and circuits.
There are several methods for solving systems of equations, each with its own strengths and weaknesses. The two most common methods are: substitution and elimination. We'll be using the substitution method today, but we'll also touch upon the elimination method to give you a broader perspective. Understanding these methods is key to becoming a proficient problem solver in mathematics.
Method 1: The Substitution Method
The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with a single variable, which we can easily solve. Once we find the value of one variable, we can substitute it back into either of the original equations to find the value of the other variable. Let's break down the steps:
Step 1: Solve one equation for one variable
Looking at our system:
2x + 3y = 12
x - y = 1
The second equation, x - y = 1, looks simpler to manipulate. Let's solve it for x:
x = y + 1
We've now expressed x in terms of y. This is the first crucial step in the substitution method. By isolating x, we've created an expression that we can substitute into the other equation. This is where the magic of substitution happens! Remember, the goal is to reduce the system to a single equation with a single variable, and we're well on our way.
Step 2: Substitute the expression into the other equation
Now, we'll substitute the expression x = y + 1 into the first equation, 2x + 3y = 12:
2(y + 1) + 3y = 12
Notice that we've replaced x with (y + 1). This is the core of the substitution method. We've effectively eliminated x from the equation, leaving us with an equation solely in terms of y. This is a major breakthrough because we can now solve for y directly.
Step 3: Solve for the remaining variable
Let's simplify and solve the equation for y:
2y + 2 + 3y = 12
5y + 2 = 12
5y = 10
y = 2
Excellent! We've found the value of y: y = 2. This is half the battle. Now that we know y, we can find x using the expression we derived earlier.
Step 4: Substitute back to find the other variable
Remember that we found x = y + 1. Now, we substitute y = 2 into this equation:
x = 2 + 1
x = 3
Fantastic! We've found the value of x: x = 3. We now have both x and y, which means we've solved the system of equations!
Step 5: Write the solution as an ordered pair
The solution to the system is the ordered pair (x, y) = (3, 2). This means that x = 3 and y = 2 satisfy both equations in the system. It's always a good idea to check your solution by plugging the values back into the original equations to make sure they hold true. This is a simple yet effective way to catch any errors you might have made along the way. Remember, accuracy is key in mathematics!
Method 2: The Elimination Method (Brief Overview)
While we solved the system using substitution, let's briefly discuss the elimination method. The elimination method involves manipulating the equations so that the coefficients of one variable are opposites. Then, you add the equations together, eliminating that variable. You can then solve for the remaining variable and substitute back to find the other one. This method is particularly useful when the coefficients of one variable are already opposites or can be easily made opposites by multiplying one or both equations by a constant. The elimination method provides another powerful tool for solving systems of equations.
For our system:
2x + 3y = 12
x - y = 1
We could multiply the second equation by -2 to get -2x + 2y = -2. Then, adding this to the first equation would eliminate x. Can you see how it works? Give it a try as an exercise!
Verification: Is Our Solution Correct?
Let's verify our solution (3, 2) by substituting x = 3 and y = 2 into the original equations:
For the first equation, 2x + 3y = 12:
2(3) + 3(2) = 6 + 6 = 12
It checks out!
For the second equation, x - y = 1:
3 - 2 = 1
It checks out too! This confirms that our solution (3, 2) is indeed correct. Always remember to verify your solutions, guys. It's a small step that can save you from making mistakes.
Why is This Important?
Solving systems of equations isn't just an abstract mathematical exercise. It's a skill that has practical applications in various fields.
- Science and Engineering: Scientists and engineers use systems of equations to model and analyze complex systems, such as electrical circuits, chemical reactions, and structural designs.
- Economics and Finance: Economists use systems of equations to model supply and demand, market equilibrium, and financial investments.
- Computer Science: Computer scientists use systems of equations in algorithms for optimization, computer graphics, and data analysis.
Understanding how to solve systems of equations provides you with a powerful problem-solving tool that can be applied across many disciplines. The ability to translate real-world problems into mathematical models and solve them is a highly valuable skill in today's world.
Conclusion
We've successfully solved the system of equations 2x + 3y = 12 and x - y = 1 using the substitution method, finding the ordered pair (3, 2). We also briefly touched upon the elimination method and highlighted the importance of verifying your solutions. Remember, practice makes perfect! The more you solve systems of equations, the more comfortable and confident you'll become. So, keep practicing, and don't hesitate to explore different methods and approaches. Math can be challenging, but it's also incredibly rewarding. Keep up the great work!
Solving systems of equations is a fundamental skill in algebra, and mastering it will serve you well in your mathematical journey. So, keep exploring, keep practicing, and keep having fun with math, guys! You've got this!