Solving Systems Of Equations By Elimination

by Chloe Fitzgerald 44 views

Hey guys! Today, we're diving deep into the world of systems of equations and tackling a common problem-solving technique: elimination. We'll break down how to use elimination to solve systems, focusing on a specific example that'll help you master this method. So, buckle up and let's get started!

Understanding Systems of Equations

Before we jump into the nitty-gritty, 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 share the same variables. The goal? To find the values of those variables that make all the equations true simultaneously. Think of it like a puzzle where each equation is a piece, and you need to find the solution that fits them all together.

Now, why do we even care about solving systems of equations? Well, they pop up all over the place in real-world scenarios! From figuring out the break-even point for a business to determining the optimal mix of ingredients in a recipe, systems of equations help us model and solve a wide range of problems. There are several methods for solving these systems, including graphing, substitution, and our focus for today: elimination.

The Power of Elimination

The elimination method, also known as the addition method, is a powerful technique for solving systems of equations by strategically manipulating the equations to eliminate one variable. This leaves us with a single equation in one variable, which is much easier to solve. Once we find the value of that variable, we can plug it back into one of the original equations to find the value of the other variable. It's like a domino effect – knock one down, and the rest fall into place!

The key to elimination lies in the idea of additive inverses. Remember those? They're pairs of numbers that add up to zero (like 5 and -5). By creating additive inverses for the coefficients of one of the variables in our system of equations, we can add the equations together and watch that variable vanish. Magic! (Well, mathematical magic, anyway.) This method is especially handy when the coefficients of one variable are multiples of each other, or when it's easy to make them multiples with a little multiplication.

Our Example System

Okay, enough theory! Let's get practical. We're going to work through a specific system of equations step-by-step to demonstrate the elimination method in action. Here's the system we'll be tackling:

4x - 9y = 7
-2x + 3y = 4

Our mission, should we choose to accept it (and we do!), is to find the values of x and y that satisfy both of these equations. Let's break down how to do it using elimination.

Eliminating the x-terms: A Step-by-Step Guide

The question at hand asks: "What number would you multiply the second equation by in order to eliminate the x-terms when adding to the first equation?" This is the core of the elimination method, guys. We need to strategically manipulate one of the equations so that when we add them together, the x variable disappears. So, how do we do it?

Identifying the Target

First, let's focus on the x-terms in our system of equations: 4x in the first equation and -2x in the second equation. To eliminate x, we need to create additive inverses. In other words, we want the coefficient of x in the second equation to become -4, so that when we add the equations, 4x + (-4x) = 0.

The Multiplication Factor

Now, what number do we need to multiply -2x by to get -4x? The answer is 2! If we multiply the entire second equation by 2, the x-term will become -4x, perfectly set up for elimination.

So, let's multiply the entire second equation (-2x + 3y = 4) by 2. Remember, we need to multiply every term in the equation to maintain the equality:

2 * (-2x + 3y) = 2 * 4
-4x + 6y = 8

The Elimination Process

Now we have a modified system of equations:

4x - 9y = 7
-4x + 6y = 8

See how the x-terms are now additive inverses (4x and -4x)? Time for the magic to happen! We add the two equations together term by term:

(4x - 9y) + (-4x + 6y) = 7 + 8

Combining like terms, we get:

0x - 3y = 15
-3y = 15

The x variable is gone! We've successfully eliminated it. Now we have a simple equation in terms of y. We can solve for y by dividing both sides by -3:

y = 15 / -3
y = -5

Finding the Value of x

We've found that y = -5. Awesome! But we're not done yet. We still need to find the value of x. To do this, we simply substitute our value of y back into either of the original equations. Let's use the first equation (4x - 9y = 7) for this:

4x - 9(-5) = 7
4x + 45 = 7

Subtract 45 from both sides:

4x = -38

Divide both sides by 4:

x = -38 / 4
x = -19 / 2
x = -9.5

The Solution

We've done it! We've solved the system of equations using elimination. Our solution is x = -9.5 and y = -5. This means that the point (-9.5, -5) is the only point that satisfies both equations in the system. You can always check your answer by plugging these values back into the original equations to make sure they hold true.

Key Takeaways and Pro Tips

So, what have we learned, guys? The elimination method is a powerful tool for solving systems of equations. Here are some key takeaways and pro tips to keep in mind:

  • Identify the Target: Look for variables with coefficients that are either the same or easy to make the same (or additive inverses) through multiplication.
  • Multiply Strategically: Multiply one or both equations by a constant that will create additive inverses for the coefficients of the variable you want to eliminate.
  • Add the Equations: Add the equations together term by term. This should eliminate one variable, leaving you with a single equation in one variable.
  • Solve for the Remaining Variable: Solve the resulting equation for the remaining variable.
  • Substitute Back: Substitute the value you found back into one of the original equations to solve for the other variable.
  • Check Your Answer: Always, always check your solution by plugging the values of x and y back into the original equations to ensure they are satisfied. This helps prevent errors and gives you confidence in your answer.
  • Be Flexible: Sometimes, you might need to multiply both equations by different constants to eliminate a variable. Don't be afraid to get creative!

Practice Makes Perfect

The best way to master the elimination method is through practice. Work through lots of examples, guys! The more you practice, the more comfortable you'll become with the steps involved, and the faster you'll be able to solve systems of equations. You can find practice problems in textbooks, online resources, or even create your own!

Conclusion

And there you have it! We've explored the elimination method for solving systems of equations, walked through a detailed example, and shared some pro tips to help you succeed. Remember, guys, solving systems of equations is a valuable skill that can be applied in many different areas. So, keep practicing, and you'll be a pro in no time!

Now go forth and conquer those equations!