Solve Systems Of Equations: Elimination Method Explained

by Henrik Larsen 57 views

Have you ever encountered a system of equations and felt completely lost on how to solve it? Don't worry, you're not alone! Many students find systems of equations challenging, but with the right approach, they can be conquered. In this article, we'll break down a common method for solving systems of equations: elimination. We'll walk through an example step-by-step, providing explanations and tips along the way. So, grab your pencil and paper, and let's dive in!

Understanding Systems of Equations

Before we tackle the solution, let's clarify what a system of equations actually is. A system of equations is a set of two or more equations that share the same variables. The goal is to find values for those variables that satisfy all equations in the system simultaneously. Think of it like finding the point where two lines intersect on a graph – that point represents the solution that works for both equations.

The Elimination Method: A Powerful Tool

The elimination method is a fantastic technique for solving systems of equations. The core idea behind this method is to manipulate the equations in such a way that either the x or y coefficients become opposites. When you add the equations together, one of the variables will be eliminated, leaving you with a single equation in one variable. This simplified equation can then be easily solved, and its solution can be substituted back into one of the original equations to find the value of the other variable. Let's get into the specifics with an example.

Example: Solving a System of Equations Using Elimination

Let's consider the following system of equations:

4x + 5y = 7
3x - 2y = -12

Our mission is to find the values of x and y that make both of these equations true. Using the elimination method, we'll follow these steps:

Step 1: Choose a Variable to Eliminate

Looking at the coefficients of x and y, we can choose to eliminate either variable. To make our lives easier, we want to pick the variable that requires the least amount of multiplication to achieve opposite coefficients. In this case, let's aim to eliminate y. The coefficients of y are 5 and -2. The least common multiple of 5 and 2 is 10, so we'll manipulate the equations to get 10y and -10y.

Step 2: Multiply the Equations

To get 10y in the first equation, we'll multiply the entire equation by 2:

2 * (4x + 5y) = 2 * 7
8x + 10y = 14

To get -10y in the second equation, we'll multiply the entire equation by 5:

5 * (3x - 2y) = 5 * (-12)
15x - 10y = -60

Now we have a modified system of equations:

8x + 10y = 14
15x - 10y = -60

Step 3: Add the Equations

Notice that the y terms now have opposite coefficients (+10 and -10). This is perfect! When we add the two equations together, the y terms will cancel out:

(8x + 10y) + (15x - 10y) = 14 + (-60)
8x + 15x + 10y - 10y = -46
23x = -46

Step 4: Solve for the Remaining Variable

We're left with a simple equation in one variable: 23x = -46. To solve for x, we divide both sides by 23:

x = -46 / 23
x = -2

Great! We've found that x = -2.

Step 5: Substitute to Find the Other Variable

Now that we know the value of x, we can substitute it back into either of the original equations to solve for y. Let's use the first equation, 4x + 5y = 7:

4 * (-2) + 5y = 7
-8 + 5y = 7

Add 8 to both sides:

5y = 15

Divide both sides by 5:

y = 3

So, we've found that y = 3.

Step 6: Check Your Solution

It's always a good idea to check your solution to make sure it works in both original equations. Let's plug x = -2 and y = 3 into the equations:

For the first equation, 4x + 5y = 7:

4 * (-2) + 5 * 3 = 7
-8 + 15 = 7
7 = 7 (Correct!)

For the second equation, 3x - 2y = -12:

3 * (-2) - 2 * 3 = -12
-6 - 6 = -12
-12 = -12 (Correct!)

Our solution works in both equations, so we can confidently say that the solution to the system is x = -2 and y = 3.

The Solution

The solution to the system of equations:

4x + 5y = 7
3x - 2y = -12

is x = -2 and y = 3. We can write this as an ordered pair: (-2, 3).

Key Takeaways and Tips for Solving Systems of Equations

  • Master the Elimination Method: This method is super versatile and can be applied to a wide variety of systems of equations. Practice makes perfect, so work through several examples to build your skills.
  • Choose the Easiest Variable to Eliminate: Look for opportunities to minimize the amount of multiplication needed. Sometimes, eliminating one variable is significantly easier than eliminating the other.
  • Don't Forget to Check Your Solution: Plugging your solution back into the original equations is a crucial step. It helps you catch any errors and ensures that your answer is correct.
  • Be Careful with Signs: Pay close attention to positive and negative signs when multiplying and adding equations. A small sign error can throw off your entire solution.
  • Practice Makes Perfect: Solving systems of equations is a skill that improves with practice. Work through various examples, and don't be afraid to ask for help if you get stuck.

Other Methods for Solving Systems of Equations

While the elimination method is a powerful tool, it's not the only method available. Two other common methods include:

  • Substitution Method: In this method, you solve one equation for one variable and substitute that expression into the other equation. This eliminates one variable and allows you to solve for the other.
  • Graphing Method: You can graph both equations on the same coordinate plane. The point where the lines intersect represents the solution to the system.

Each method has its strengths and weaknesses, and the best method to use often depends on the specific system of equations you're dealing with.

Conclusion

Solving systems of equations might seem daunting at first, but with a systematic approach and plenty of practice, you can become a pro! The elimination method, as we've demonstrated, is a highly effective technique. Remember to choose the variable that's easiest to eliminate, multiply the equations carefully, add them together to eliminate a variable, solve for the remaining variable, and finally, substitute back to find the other variable. And always, always check your solution!

So, there you have it! Go forth and conquer those systems of equations, guys! You've got this!