Solving Systems Of Equations By Addition Method
Hey everyone! Today, let's dive into a super important concept in algebra solving systems of equations using the addition method. It might sound intimidating, but trust me, it's actually pretty straightforward once you get the hang of it. We'll break it down step by step, and by the end of this article, you'll be solving these like a pro!
What are Systems of Equations?
Before we jump into the addition method, let's quickly recap what systems of equations are all about. Systems of equations are basically a set of two or more equations that share the same variables. Our goal is to find the values for those variables that make all the equations in the system true at the same time. Think of it like finding the sweet spot that satisfies all the conditions.
For example, we might have two equations like:
- x + y = 10
- x - y = 2
Here, we have two equations with two variables, x and y. The solution to this system would be the values of x and y that make both of these equations true. There are several ways to solve such systems, and today we are focusing on the addition method.
The Addition Method Unveiled
The addition method, also known as the elimination method, is a technique used to solve systems of equations by adding the equations together in a way that eliminates one of the variables. This leaves us with a single equation with just one variable, which we can easily solve. Once we find the value of one variable, we can substitute it back into one of the original equations to find the value of the other variable. Cool, right?
Step-by-Step Guide to the Addition Method
Let's walk through the steps of the addition method with a concrete example. Suppose we have the following system of equations:
- 2x + y = 8
- x - y = 1
Ready? Let's get started:
Step 1: Align the Equations
The first thing we need to do is make sure the equations are lined up nicely. This means that the x terms, the y terms, and the constant terms should be in the same columns. In our example, the equations are already aligned perfectly:
2x + y = 8
x - y = 1
Step 2: Check for Opposing Coefficients
This is the crucial step where we look for a pair of terms (either the x terms or the y terms) that have opposite coefficients. Opposite coefficients are numbers that have the same absolute value but different signs (e.g., 3 and -3, or 1 and -1).
In our system, notice that the y terms have coefficients of +1 and -1. Bingo! This is exactly what we want. If we add the equations together, the y terms will cancel out. If you don't have opposing coefficients, don't worry! We'll cover how to handle that in the next step.
Step 3: Multiply (if necessary)
What if we don't have opposing coefficients right away? No problem! We can multiply one or both of the equations by a constant to create them. The goal here is to find a number that, when multiplied by one of the coefficients, will give us the opposite of the other coefficient.
Let's say we had the following system:
- 3x + 2y = 11
- x + y = 4
In this case, we don't have opposing coefficients. But we can easily create them. Notice that if we multiply the second equation by -2, the y term will become -2y, which is the opposite of the 2y in the first equation.
So, let's multiply the entire second equation by -2:
-2 * (x + y) = -2 * 4
-2x - 2y = -8
Now our system looks like this:
- 3x + 2y = 11
- -2x - 2y = -8
Now we have opposing coefficients for the y terms!
Step 4: Add the Equations
Now for the fun part! We're going to add the equations together, column by column. This means we'll add the x terms, the y terms, and the constant terms separately.
Going back to our original example:
2x + y = 8
x - y = 1
Let's add them up:
(2x + x) + (y + (-y)) = 8 + 1
3x + 0 = 9
3x = 9
Notice how the y terms canceled out, leaving us with a simple equation with just x.
Step 5: Solve for the Remaining Variable
Now we have a single equation with one variable. We can easily solve for x by dividing both sides by 3:
3x = 9
x = 9 / 3
x = 3
Great! We've found the value of x. It's 3.
Step 6: Substitute to Find the Other Variable
We're not done yet! We still need to find the value of 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 one you choose; you'll get the same answer either way.
Let's use the second equation, x - y = 1:
3 - y = 1
Now we can solve for y. Subtract 3 from both sides:
-y = 1 - 3
-y = -2
Multiply both sides by -1 to get y by itself:
y = 2
Alright! We've found the value of y. It's 2.
Step 7: Check Your Solution
It's always a good idea to check your solution to make sure it's correct. To do this, we'll substitute the values of x and y into both of the original equations and see if they hold true.
Let's check with our solution, x = 3 and y = 2:
- Equation 1: 2x + y = 8
- 2(3) + 2 = 8
- 6 + 2 = 8
- 8 = 8 (True!)
- Equation 2: x - y = 1
- 3 - 2 = 1
- 1 = 1 (True!)
Since our solution satisfies both equations, we know it's correct.
Step 8: Write the Solution as an Ordered Pair
Finally, we write our solution as an ordered pair (x, y). In our case, the solution is (3, 2).
And that's it! We've successfully solved the system of equations using the addition method. Give yourself a pat on the back!
Answering the Question: x - y = 5 and x + y = 7
Okay, guys, let's tackle the specific question you posed earlier: What's the solution to the system of equations using the addition method, considering the equations x - y = 5 and x + y = 7? And we need to pick the correct option from:
A) (6, 1) B) (5, 2) C) (4, -1) D) (3, 2)
Let's follow our step-by-step guide to solve this:
Step 1: Align the Equations
Our equations are already nicely aligned:
x - y = 5
x + y = 7
Step 2: Check for Opposing Coefficients
Look at that! The y terms have coefficients of -1 and +1. Perfect! We have opposing coefficients.
Step 3: Multiply (if necessary)
No need to multiply in this case. We already have opposing coefficients.
Step 4: Add the Equations
Let's add the equations together:
(x + x) + (-y + y) = 5 + 7
2x + 0 = 12
2x = 12
Step 5: Solve for the Remaining Variable
Divide both sides by 2 to solve for x:
2x = 12
x = 12 / 2
x = 6
So, x = 6.
Step 6: Substitute to Find the Other Variable
Let's substitute x = 6 into the second equation, x + y = 7:
6 + y = 7
Subtract 6 from both sides to solve for y:
y = 7 - 6
y = 1
So, y = 1.
Step 7: Check Your Solution
Let's check our solution, x = 6 and y = 1, in both equations:
- Equation 1: x - y = 5
- 6 - 1 = 5
- 5 = 5 (True!)
- Equation 2: x + y = 7
- 6 + 1 = 7
- 7 = 7 (True!)
Our solution checks out.
Step 8: Write the Solution as an Ordered Pair
The solution is (6, 1).
So, the correct answer is A) (6, 1).
Justification:
We solved the system of equations x - y = 5 and x + y = 7 using the addition method. By adding the two equations, we eliminated the variable y and solved for x. Then, we substituted the value of x back into one of the original equations to solve for y. This gave us the solution (x, y) = (6, 1), which is option A.
Practice Makes Perfect
The addition method might seem tricky at first, but the more you practice, the easier it will become. Try solving different systems of equations using this method, and you'll soon be a pro. Remember, the key is to line up the equations, look for opposing coefficients (or create them), add the equations, solve for one variable, substitute to find the other, and always check your solution.
Conclusion
Alright, guys! We've covered the addition method for solving systems of equations. You now have a powerful tool in your algebra arsenal. Keep practicing, and you'll be amazed at how easily you can solve these problems. Remember, math can be fun, especially when you understand the concepts. Keep exploring, keep learning, and keep solving!
