Solving X+y=12 And 2x-y=-3 A Step-by-Step Guide

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Are you struggling with systems of equations? Don't worry, guys! You're not alone. Many students find these problems tricky, but with a clear, step-by-step approach, you can conquer them. In this guide, we'll break down how to solve the system of equations x + y = 12 and 2x - y = -3. We'll use the elimination method, which is a super effective way to tackle these types of problems. So, grab your pencil and paper, and let's dive in!

Understanding Systems of Equations

Before we jump into solving this specific problem, let's make sure we're all on the same page about 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 the equations in the system simultaneously. In other words, we want to find the values of x and y that make both x + y = 12 and 2x - y = -3 true.

Think of it like this: each equation represents a relationship between the variables. We're looking for the point where those relationships intersect. Graphically, each equation represents a line, and the solution to the system is the point where the lines cross. But we're not going to graph them today; we're going to use algebra to find that point.

The system of equations we're tackling today, x + y = 12 and 2x - y = -3, is a classic example. It's got two equations and two unknowns (x and y), which means we can use various methods to solve it. We could use substitution, where we solve one equation for one variable and plug it into the other equation. But for this particular system, the elimination method is going to be our best friend. It's often the quickest and most straightforward way to solve systems where the coefficients of one of the variables are opposites or easily made into opposites.

Choosing the Elimination Method

So, why are we choosing the elimination method for this particular system of equations? Well, take a look at our equations: x + y = 12 and 2x - y = -3. Notice anything special about the 'y' terms? One has a positive sign (+y) and the other has a negative sign (-y). This is a huge clue that the elimination method is the way to go. The elimination method works best when you can easily add or subtract the equations to eliminate one of the variables.

The beauty of the elimination method is that it simplifies the problem. By strategically adding or subtracting the equations, we can get rid of one variable and end up with a single equation in just one variable. This is much easier to solve! Once we've solved for one variable, we can plug that value back into either of the original equations to find the value of the other variable. It's like a domino effect – solve for one, then the other falls into place.

Now, you might be wondering, what if the coefficients weren't opposites? What if we had something like 2y and 3y? No problem! We can still use the elimination method. We would just need to multiply one or both equations by a constant so that the coefficients of one of the variables become opposites. We'll see an example of this later. But for this system, we're in luck – the 'y' terms are perfectly set up for elimination.

Step 1: Align the Equations

Before we can start eliminating variables, it's crucial to align the equations. This means writing the equations one above the other, making sure that the x terms, y terms, and constant terms are lined up in columns. This makes the next step – adding the equations – much easier and helps prevent careless errors. Think of it like organizing your workspace before you start a project – a little preparation goes a long way!

So, let's write our equations neatly, one above the other:

x + y = 12
2x - y = -3

See how the 'x' terms are in a column, the 'y' terms are in a column, and the constants are in a column? This neat arrangement is key to the success of the elimination method. It's like making sure all the pieces of a puzzle are facing the right way before you try to fit them together. Now that our equations are aligned, we're ready for the next step: adding the equations.

Step 2: Add the Equations

This is where the magic happens! Now that our equations are neatly aligned, we can add the equations together. We're going to add the left-hand sides of the equations and the right-hand sides of the equations separately. The goal here is to eliminate one of the variables. Remember how we noticed that the 'y' terms had opposite signs? This is why this step is going to be so effective.

Let's add the equations:

 (x + y) + (2x - y) = 12 + (-3)

Now, let's simplify. On the left side, we have x + 2x, which is 3x. And then we have y - y, which is 0. The 'y' terms have been eliminated! On the right side, we have 12 + (-3), which is 9. So, our equation becomes:

3x = 9

See how adding the equations eliminated the 'y' variable and left us with a simple equation in just 'x'? This is the power of the elimination method! We've reduced the problem to a single equation that we can easily solve. It's like simplifying a complex recipe down to its essential ingredients. Now, let's solve for 'x'.

Step 3: Solve for x

We've got a simple equation: 3x = 9. To solve for x, we need to isolate 'x' on one side of the equation. This means getting rid of the 3 that's multiplying 'x'. How do we do that? We divide both sides of the equation by 3. Remember, whatever we do to one side of the equation, we have to do to the other side to keep it balanced. It's like a seesaw – if you add weight to one side, you need to add the same weight to the other side to keep it level.

So, let's divide both sides of 3x = 9 by 3:

3x / 3 = 9 / 3

This simplifies to:

x = 3

Fantastic! We've found the value of x. It's 3. We're halfway there! Now that we know the value of one variable, we can use it to find the value of the other variable. It's like finding one piece of a puzzle – it helps you see how the other pieces fit.

Step 4: Solve for y

Now that we know x = 3, we can solve for y. To do this, we'll substitute the value of x (which is 3) into either of the original equations. It doesn't matter which equation we choose; we'll get the same answer for y. But to keep things simple, let's choose the first equation, x + y = 12. It looks a little easier to work with.

So, let's substitute x = 3 into x + y = 12:

3 + y = 12

Now we have a simple equation with just one variable, y. To solve for y, we need to isolate y on one side of the equation. This means getting rid of the 3 that's being added to y. We do this by subtracting 3 from both sides of the equation:

3 + y - 3 = 12 - 3

This simplifies to:

y = 9

Excellent! We've found the value of y. It's 9. We've solved for both x and y! It's like finding the last piece of the puzzle and completing the picture.

Step 5: Check Your Solution

Before we celebrate our victory, it's always a good idea to check our solution. This ensures that we haven't made any mistakes along the way. To check our solution, we'll substitute the values we found for x and y (x = 3 and y = 9) into both of the original equations. If both equations are true, then our solution is correct.

Let's start with the first equation, x + y = 12. Substituting x = 3 and y = 9, we get:

3 + 9 = 12

This is true! Now let's check the second equation, 2x - y = -3. Substituting x = 3 and y = 9, we get:

2(3) - 9 = -3
6 - 9 = -3
-3 = -3

This is also true! Since our values for x and y satisfy both equations, we can be confident that our solution is correct.

The Solution

We've done it! We've successfully solved the system of equations x + y = 12 and 2x - y = -3. The solution is x = 3 and y = 9. We can write this as an ordered pair (3, 9). This means that the point (3, 9) is the point where the lines represented by these two equations intersect on a graph.

Congratulations! You've mastered the elimination method and solved a system of equations. Remember, the key is to break the problem down into smaller, manageable steps. And always check your solution to make sure you're on the right track.

Practice Makes Perfect

Now that you've seen how to solve this system of equations, the best way to solidify your understanding is to practice. Try solving similar problems on your own. You can find plenty of examples in textbooks, online resources, or even by making up your own systems of equations.

Here are a few tips to keep in mind as you practice:

  • Look for opportunities to use the elimination method. If the coefficients of one of the variables are opposites or easily made into opposites, elimination is often the quickest approach.
  • Don't be afraid to multiply equations. Sometimes you'll need to multiply one or both equations by a constant to make the coefficients of a variable match up before you can eliminate them.
  • Stay organized. Write your equations neatly and keep your work organized to avoid careless errors.
  • Check your solutions. Always substitute your values for x and y back into the original equations to make sure they work.

Solving systems of equations is a valuable skill in mathematics and many real-world applications. With practice and a clear understanding of the methods, you'll be able to tackle these problems with confidence. So keep practicing, and you'll become a systems of equations pro in no time!

Beyond the Basics: Other Methods and Applications

While we've focused on the elimination method in this guide, it's important to know that there are other ways to solve systems of equations. One common method is substitution, where you solve one equation for one variable and substitute that expression into the other equation. Substitution is often a good choice when one of the equations is already solved for one variable or can be easily solved.

Another method is graphing. As we mentioned earlier, each equation in a system represents a line, and the solution to the system is the point where the lines intersect. Graphing can be a helpful visual way to understand systems of equations, especially for simple systems with integer solutions.

Systems of equations aren't just a math textbook concept; they have applications in many fields. They can be used to model and solve problems in science, engineering, economics, and more. For example, they can be used to:

  • Determine the break-even point for a business
  • Calculate the optimal mix of ingredients in a recipe
  • Predict the trajectory of a projectile
  • Analyze supply and demand in economics

Understanding systems of equations opens up a world of possibilities for problem-solving. So keep exploring, keep learning, and keep applying your knowledge!

Final Thoughts

Solving systems of equations can seem daunting at first, but with a systematic approach and plenty of practice, it becomes a manageable and even enjoyable task. We've walked through the elimination method step by step, and hopefully, you now feel more confident in your ability to solve these types of problems. Remember to align your equations, choose the best method (elimination, substitution, or graphing), solve for one variable at a time, and always check your solution. Keep practicing, and you'll be a system of equations master in no time! You got this, guys!