Solve X+y=12, X-y=6 By Addition Method

Hey guys! Today, we're diving into solving a system of equations using the addition method. It's a super handy technique, especially when you've got equations lined up just right. Our system is:

1. x + y = 12
2. x - y = 6

Let's break it down step by step so you can master this method!

Understanding the Addition Method

The addition method, also known as the elimination method, works by adding two equations together in a way that one of the variables cancels out. This leaves you with a single equation with one variable, which is much easier to solve. The beauty of this method lies in its simplicity and effectiveness, particularly when dealing with linear equations. When you're faced with a system of equations, it’s like having a puzzle. The addition method provides a systematic approach to crack that puzzle. You're essentially manipulating the equations to reveal the values of the unknowns. This method is especially useful when the coefficients of one variable are opposites or can be easily made opposites through multiplication. By eliminating one variable, you simplify the problem, making it more manageable and straightforward to solve. The core idea is to strategically combine the equations to isolate one variable, making the solution process more efficient. Understanding this fundamental principle is key to mastering the addition method and applying it effectively to various systems of equations.

Step 1: Align the Equations

The first thing we need to do is make sure our equations are aligned. This means having the x terms, y terms, and constants lined up in columns. Luckily, our equations are already set up perfectly:

x + y = 12 x - y = 6

When dealing with systems of equations, proper alignment is crucial for the addition method to work effectively. This involves arranging the equations so that like terms are vertically aligned. Specifically, the 'x' terms should be in one column, the 'y' terms in another, and the constant terms on the other side of the equals sign. This alignment ensures that when you add the equations together, you are only combining terms that have the same variable. For example, if the equations were not initially aligned, you might need to rearrange them. Suppose you have x + y = 12 and -y + x = 6. You would rewrite the second equation as x - y = 6 to achieve proper alignment. Without this alignment, adding the equations could lead to confusion and incorrect results. Proper alignment sets the stage for the next step, where you add the equations together, making the elimination of one variable a seamless and accurate process. Therefore, always double-check that your equations are correctly aligned before proceeding with the addition method.

Step 2: Add the Equations

Now, we add the two equations together. Notice that the +y and -y terms will cancel each other out:

(x + y) + (x - y) = 12 + 6

x + y + x - y = 18

2x = 18

Adding the equations is the heart of the addition method. As you combine the equations, focus on how the terms align and interact. In our case, adding the left-hand sides (x + y) and (x - y) and the right-hand sides (12 and 6) simplifies the system. The goal is to have one variable eliminated, making the resulting equation easier to solve. When adding the equations, pay close attention to the signs of the terms. For example, if we had x + y = 12 and -x + 2y = 6, adding them would result in (x - x) + (y + 2y) = 12 + 6, which simplifies to 3y = 18. In our original problem, the 'y' terms cancel each other out because they have opposite signs but the same coefficient. This cancellation results in an equation with only one variable, 'x', which we can then solve directly. Accurate addition is essential to ensure the correct elimination of a variable and the correct simplification of the equation. By carefully combining the terms, you pave the way for a straightforward solution to the system of equations.

Step 3: Solve for x

We've got 2x = 18. To solve for x, we divide both sides by 2:

2x / 2 = 18 / 2

x = 9

Solving for 'x' is a crucial step in finding the complete solution to the system of equations. Once we've simplified the equation to 2x = 18, we isolate 'x' by dividing both sides of the equation by the coefficient of 'x', which in this case is 2. This process ensures that we maintain the equality of the equation while isolating 'x' on one side. By dividing both sides by 2, we get x = 9. This value represents the 'x'-coordinate of the solution to the system. It's important to perform the division accurately to avoid errors. Double-checking your work at this stage can help ensure that you have the correct value for 'x'. With 'x' now determined, we can move on to the next step: substituting 'x' back into one of the original equations to find the value of 'y'. This systematic approach helps us find the values of both variables, providing the complete solution to the system of equations. Therefore, accurately solving for 'x' is a key milestone in the process.

Step 4: Substitute x into One of the Original Equations

Now that we know x = 9, we can plug it into either of the original equations to solve for y. Let's use the first equation, x + y = 12:

9 + y = 12

Substituting the value of 'x' into one of the original equations is a pivotal step in determining the value of 'y'. This process allows us to use the known value of 'x' to solve for the remaining unknown variable. By replacing 'x' with 9 in the equation x + y = 12, we create a new equation that contains only 'y' as the variable. This substitution transforms the equation into a simple algebraic expression that is easy to solve. It's important to choose an equation that is straightforward to work with, which can sometimes depend on the specific system of equations. For example, if one equation has smaller coefficients or fewer terms, it might be easier to use for substitution. Once the substitution is made, we can proceed to isolate 'y' and find its value. This step brings us closer to the complete solution, as we now have values for both 'x' and 'y'. Accurate substitution is crucial to avoid errors, so carefully replace 'x' with its numerical value and double-check your work before moving on to the next step.

Step 5: Solve for y

To solve for y, subtract 9 from both sides:

9 + y - 9 = 12 - 9

y = 3

Solving for 'y' involves isolating 'y' on one side of the equation to determine its value. After substituting x = 9 into the equation x + y = 12, we have 9 + y = 12. To isolate 'y', we subtract 9 from both sides of the equation. This maintains the balance of the equation while moving the constant term to the other side. By subtracting 9 from both sides, we get y = 12 - 9, which simplifies to y = 3. This value represents the 'y'-coordinate of the solution to the system of equations. It's important to perform the subtraction accurately to ensure that the value of 'y' is correct. Once we have solved for 'y', we have both the 'x' and 'y' values, which together form the solution to the system. This solution represents the point where the two lines described by the equations intersect on a graph. Double-checking your calculations is always a good idea to confirm the accuracy of your results. With both 'x' and 'y' values determined, we can confidently state the solution to the system of equations.

Step 6: Check Your Solution

To make sure our solution is correct, we can plug x = 9 and y = 3 into both original equations:

1. x + y = 12 -> 9 + 3 = 12 -> 12 = 12 (Correct!)
2. x - y = 6 -> 9 - 3 = 6 -> 6 = 6 (Correct!)

Checking the solution is a critical step to ensure the accuracy of our calculations and the validity of our results. This involves substituting the values we found for 'x' and 'y' back into the original equations to verify that they satisfy both equations. By plugging x = 9 and y = 3 into the first equation, x + y = 12, we get 9 + 3 = 12, which simplifies to 12 = 12. This confirms that the first equation holds true with our solution. Similarly, substituting the values into the second equation, x - y = 6, we get 9 - 3 = 6, which simplifies to 6 = 6. This confirms that the second equation also holds true. If both equations are satisfied, we can confidently conclude that our solution is correct. However, if either equation does not hold true, it indicates that there was an error in our calculations, and we need to go back and review our steps. Checking the solution provides peace of mind and ensures that we have accurately solved the system of equations. Therefore, always make it a habit to verify your results before finalizing your answer.

Final Answer

The solution to the system of equations is x = 9 and y = 3. So the final answer is (9, 3).

Therefore, the final answer is: x = 9, y = 3

Hope this helps you guys out! Let me know if you have any other questions. Keep practicing, and you'll get the hang of it in no time!