Solving Equations: Elimination By Addition

Hey everyone! Today, we're diving into a cool method for solving systems of equations: the elimination-by-addition method. This is a super handy technique, especially when you're dealing with equations that seem a little tricky at first glance. We will be using the following equations: $x - 2y = 5$ and $5x - y = -2$. This method is all about strategically manipulating our equations to eliminate one of the variables, making it easier to find the values of x and y.

First off, let's get a good grasp of what we're working with. A system of equations is simply a set of two or more equations that we're trying to solve simultaneously. In other words, we want to find the values of the variables (in this case, x and y) that satisfy all the equations in the system. The elimination-by-addition method is one of the most popular ways to do this, and it's pretty straightforward once you get the hang of it. The basic idea is to multiply one or both equations by a constant so that when you add the equations together, either the x or y terms cancel out. This leaves you with a single equation with just one variable, which is easy to solve. Let's break down the process step by step, making sure you're following along and feeling confident at each stage. Don't worry if it seems a little confusing at first; practice makes perfect! By the end of this guide, you will be eliminating variables like a math pro! We'll take it slow, explain all the whys and hows, and even throw in some helpful tips to make sure you nail it. Get ready to flex those equation-solving muscles!

Step-by-Step Breakdown

Now, let's get down to brass tacks and solve the system: $x - 2y = 5$ and $5x - y = -2$.

Step 1: Choose a Variable to Eliminate

Alright, the first thing we need to do is decide which variable we want to eliminate. We can choose to eliminate x or y. There's no right or wrong choice here; it often comes down to what looks easier to manipulate. In this case, let's go with eliminating y. Our goal is to get the coefficients of y to be opposites so that they cancel out when we add the equations. It means we want the y terms to have the same number, but with opposite signs (e.g., +2y and -2y). So that when we add the equations together, those terms vanish, leaving us with an equation with just x. Thinking about how to get the coefficients to cancel out is a key part of this step. For instance, if we want to eliminate y, we need to make the coefficients of y in both equations opposites. Take a look at the equations again: $x - 2y = 5$ and $5x - y = -2$. The first equation has -2y, and the second has -y. To get the coefficients of y to be opposites, we need to multiply the second equation by -2. This means everything in that equation gets multiplied by -2. Remember, the goal is to make the coefficients of one of the variables cancel out when the equations are added. This strategic move is the foundation of the elimination method! Keep in mind that you can always choose to eliminate x instead. The steps will be slightly different, but the final answer will be the same. The flexibility of this method is one of its biggest advantages. You get to decide which variable to tackle first.

Step 2: Manipulate the Equations

Okay, now that we've decided to eliminate y, we need to manipulate our equations so that the coefficients of y are opposites. As we said, we will be multiplying the second equation by -2. So, here's how that looks:

• Original second equation: 5x - y = -2
• Multiply by -2: -2 * (5x - y) = -2 * (-2)
• Resulting equation: -10x + 2y = 4

Now our system of equations looks like this:

• Equation 1: x - 2y = 5
• Equation 2 (modified): -10x + 2y = 4

Notice how the y terms now have opposite coefficients (-2y and +2y). This means they're ready to cancel out when we add the equations. This step is crucial because it sets the stage for eliminating one of the variables. Without this manipulation, we wouldn't be able to simplify the system and solve for the variables. Make sure to distribute the multiplication correctly across all terms in the equation. This is a common area where mistakes can happen, so double-check your work. Also, remember that when you multiply an equation by a number, you must multiply every term on both sides of the equals sign. This maintains the equality of the equation.

Step 3: Add the Equations

Alright, now we're ready to add our two equations together. Let's write them down again, one above the other, and add the corresponding terms:

x - 2y = 5
-10x + 2y = 4

Adding the terms together, we get:

• (x + (-10x)) + (-2y + 2y) = 5 + 4
• -9x + 0y = 9
• -9x = 9

Notice how the y terms neatly canceled out, leaving us with an equation with just x. This is exactly what we wanted! The y variable is gone. We're one step closer to our answer. Adding the equations is a simple matter of combining like terms. Make sure to add the x terms together, the y terms together, and the constants together. The key is to do this systematically and carefully. Once you get the hang of it, adding equations becomes second nature. This is where you really start to see the power of the elimination method. By adding the equations, we've simplified our system into a much more manageable form. You're now well on your way to solving for x. It's time to find the final value of x.

Step 4: Solve for the Remaining Variable

We've got a new equation: -9x = 9. Now, this is easy to solve for x. To isolate x, we need to divide both sides of the equation by -9:

• -9x / -9 = 9 / -9
• x = -1

Voila! We've found the value of x! It's -1. This is the first piece of our solution. Now you see, solving for x is a straightforward process once you've eliminated one of the variables. It's all about isolating the variable and performing the necessary arithmetic operations. Remember to perform the same operation on both sides of the equation to maintain equality. And always double-check your calculations to avoid any silly mistakes. Now that we know x = -1, we're halfway there! We still need to find the value of y. To get that, we will be replacing x in one of the original equations. Let's keep going and find y.

Step 5: Substitute to Find the Other Variable

Now that we know x = -1, we can substitute this value back into one of the original equations to solve for y. Let's use the first equation: x - 2y = 5. Replacing x with -1, we get:

• -1 - 2y = 5

Now, we need to solve for y. Let's go step by step:

1. Add 1 to both sides: -2y = 6
2. Divide both sides by -2: y = -3

And there you have it! We've found that y = -3. This step is crucial because it allows us to find the value of the other variable in the system. Substituting the known value back into one of the original equations is a simple yet powerful technique. Remember, you can choose either of the original equations. The result will be the same. Always make sure to perform the substitution and solve for the remaining variable carefully. Now we have both values.

Step 6: Write the Solution

We've found that x = -1 and y = -3. The solution to the system of equations is therefore the ordered pair (-1, -3). This means that the point (-1, -3) is the intersection point of the two lines represented by the equations. This ordered pair is the solution to the system because when you plug these values into both of the original equations, they satisfy both equations simultaneously. The solution is written as an ordered pair because, in a coordinate system, each solution is a specific point. This point satisfies all the equations.

Key Takeaways and Tips

Let's quickly recap what we've learned and go over some important tips:

• The Goal: The main idea is to eliminate one variable by adding the equations together.
• Strategic Manipulation: Multiply one or both equations by constants to make the coefficients of one variable opposites.
• Careful Addition: Add the equations carefully, term by term.
• Substitution: Substitute the value of the solved variable back into one of the original equations to find the other variable.
• Check Your Work: Always check your answer by substituting the values of x and y back into both original equations. If they satisfy both equations, you know your solution is correct!

This process might seem like a lot at first, but with practice, it becomes second nature. The elimination-by-addition method is incredibly versatile and can be applied to various types of systems of equations. Keep practicing, and you'll become a pro in no time! If you struggle with the steps, go back and review the steps or revisit the examples. Math can be tricky sometimes, but don't get discouraged. Always be patient with yourself, and celebrate your progress! Keep up the great work, and you'll be acing these equations in no time!

Conclusion

Great job, everyone! You've successfully navigated the elimination-by-addition method. Remember to practice these steps to solidify your understanding. With consistent practice, you'll become adept at solving systems of equations with ease. Keep learning, keep practicing, and never be afraid to ask for help. You got this!