Solve Equations By Elimination: Step-by-Step Guide

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Hey guys! Today, we're diving deep into the world of solving systems of equations using the elimination method. This method is super useful when you have two equations with two variables, and you want to find the values of those variables that satisfy both equations simultaneously. We'll walk through a specific example step-by-step, so you can see exactly how it's done. Let's get started!

Understanding the Elimination Method

Before we jump into the problem, let's quickly recap what the elimination method is all about. The main goal in the elimination method is to manipulate the equations in such a way that when you add them together, one of the variables cancels out. This leaves you with a single equation in a single variable, which is much easier to solve. Once you've found the value of one variable, you can plug it back into either of the original equations to find the value of the other variable. It's like a mathematical magic trick, but totally logical and reliable!

To effectively use the elimination method, it's essential to identify which variable is easiest to eliminate. This often involves looking at the coefficients of the variables in both equations. If you see that the coefficients of one variable are multiples of each other (or can easily be made into multiples), then that's a good candidate for elimination. Additionally, if the coefficients have opposite signs, that's even better, because you might be able to eliminate the variable just by adding the equations together without any initial multiplication. Recognizing these patterns is key to choosing the most efficient path to the solution. So, keep your eyes peeled for these opportunities, and you'll become a master of elimination in no time!

When we talk about solving systems of equations, we're really talking about finding the point (or points) where the lines represented by those equations intersect. In a two-variable system, each equation represents a line on a graph. The solution to the system is the point where these lines cross each other. This point satisfies both equations, meaning its x and y coordinates make both equations true. Graphing the equations can give you a visual understanding of the solution, but the elimination method gives us a precise algebraic way to find it. Sometimes the lines might be parallel, meaning they never intersect, and in that case, the system has no solution. Other times, the lines might be the same line, meaning they intersect at every point, and the system has infinitely many solutions. The elimination method helps us uncover these scenarios too.

The Problem at Hand

Okay, let's dive into our specific problem. We're given the following system of equations:

8x + 7y = 39
4x - 14y = -68

Our mission, should we choose to accept it (and we do!), is to solve this system using the elimination method. The instructions tell us to first multiply the first equation to enable the elimination of the y-term. So, that's exactly what we'll do.

Step 1: Multiplying the First Equation

To eliminate the y-term, we need to make the coefficients of y in both equations opposites of each other. Currently, we have 7y in the first equation and -14y in the second equation. Notice that 14 is a multiple of 7. Specifically, 14 is 7 times 2. So, if we multiply the first equation by 2, the coefficient of y will become 14. But to get opposites, we actually need one to be 14 and the other to be -14, which we already have. So, let's multiply the entire first equation by 2:

2 * (8x + 7y) = 2 * 39

This gives us:

16x + 14y = 78

Now we have a new, equivalent system of equations:

16x + 14y = 78
4x - 14y = -68

See how the y-terms are now set up perfectly for elimination? One has a +14 coefficient, and the other has a -14 coefficient. This is exactly what we wanted!

Step 2: Adding the Equations

Now comes the fun part: adding the equations together. When we add the equations, we add the left-hand sides together and the right-hand sides together. This is valid because if two things are equal (the left-hand side equals the right-hand side in each equation), then adding those equal things together will still result in equal things.

So, let's add the equations:

(16x + 14y) + (4x - 14y) = 78 + (-68)

Combining like terms, we get:

16x + 4x + 14y - 14y = 78 - 68
20x = 10

Look at that! The y-terms have canceled out, just like we planned. We're left with a simple equation in x.

Step 3: Solving for x

Now, let's solve for x. We have:

20x = 10

To isolate x, we divide both sides of the equation by 20:

x = 10 / 20
x = 1/2

So, we've found the value of x! x equals 1/2. That's one variable down, one to go.

Step 4: Solving for y

To find the value of y, we can substitute the value of x we just found (which is 1/2) into either of the original equations. It doesn't matter which one we choose; we should get the same value for y either way. For simplicity, let's use the first original equation:

8x + 7y = 39

Substitute x = 1/2:

8 * (1/2) + 7y = 39
4 + 7y = 39

Now, we need to isolate y. First, subtract 4 from both sides:

7y = 39 - 4
7y = 35

Finally, divide both sides by 7:

y = 35 / 7
y = 5

Alright! We've found the value of y, which is 5.

Step 5: Checking the Solution

It's always a good idea to check our solution to make sure we haven't made any mistakes. To do this, we substitute the values of x and y we found (x = 1/2 and y = 5) into both original equations and see if they hold true.

Let's check the first equation:

8x + 7y = 39
8 * (1/2) + 7 * 5 = 39
4 + 35 = 39
39 = 39

That checks out! Now let's check the second equation:

4x - 14y = -68
4 * (1/2) - 14 * 5 = -68
2 - 70 = -68
-68 = -68

That checks out too! So, our solution is correct.

The Solution

We've successfully solved the system of equations! The solution is x = 1/2 and y = 5. We can write this as an ordered pair (1/2, 5), which represents the point where the two lines intersect on a graph.

Conclusion

And there you have it, guys! We've walked through a complete example of solving a system of equations using the elimination method. We saw how to manipulate the equations by multiplying them, how to add the equations to eliminate a variable, and how to solve for the remaining variable. We also learned the crucial step of checking our solution to ensure accuracy.

The elimination method is a powerful tool in your mathematical arsenal, and with practice, you'll become a pro at using it. Keep practicing, and you'll be solving systems of equations like a boss! Remember, math isn't about memorizing formulas; it's about understanding the logic and applying it. So, keep exploring, keep questioning, and most importantly, keep having fun with math!