Solving Linear Equations: Elimination Method

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Hey guys! Let's dive into the awesome world of solving systems of linear equations using the elimination method. It's a super handy technique that helps us find the values of x and y (or whatever variables we've got) that satisfy all the equations in the system. We're going to take a look at the elimination method, which is a fantastic way to crack these problems. This method is all about strategically manipulating the equations to eliminate one of the variables, allowing us to solve for the other. Once we have that value, we can easily plug it back into one of the original equations to find the remaining variable. It's like a mathematical puzzle, and we're the detectives figuring out the solution! Let's get started and make sure that we have a solid understanding of it. We'll focus on how to use elimination, so you can ace these types of problems. Let's start with the example. -2x + 5y = -15 and 5x + 2y = -6. The question is How could you solve this system using elimination? Choose two correct answers.

Understanding the Elimination Method

So, what's the elimination method all about? The basic idea is to manipulate the equations in a system so that when you add or subtract them, one of the variables cancels out. This leaves you with a single equation in a single variable, which is super easy to solve. The key is to make the coefficients (the numbers in front of the variables) of one of the variables opposites. This way, when you add the equations, those terms disappear, and you can solve for the remaining variable. This is where the real fun begins. The elimination method is all about strategically manipulating the equations to eliminate one of the variables. Then we solve the other variable, and we can easily plug it back into one of the original equations to find the remaining variable. So think of this as a mathematical puzzle, and we're the detectives figuring out the solution! Now let's get into the nitty-gritty and see how it works with our example! Let's go back to the questions: -2x + 5y = -15 and 5x + 2y = -6. The question is How could you solve this system using elimination? Choose two correct answers. The most important thing about the elimination method is to pick the two correct answers from the question.

Let's break it down step by step and choose the answers to the question. First, take a look at our equations again:

  • -2x + 5y = -15
  • 5x + 2y = -6

We want to eliminate either x or y. There are several ways to do this, and the most efficient way depends on the specific coefficients in the equations. Here's how the options play out, so you can better understand how to solve this equation using elimination. The cool thing about this method is that there can be different approaches depending on the numbers in your equations. We can get to the heart of the elimination method with this equation. Let's use this method to solve the problem in a step-by-step manner.

Option 1: Multiply the first equation by 2 and the second equation by 5, then subtract.

If we multiply the first equation by 2 and the second equation by 5, we get:

  • 2 * (-2x + 5y) = 2 * (-15) which simplifies to -4x + 10y = -30
  • 5 * (5x + 2y) = 5 * (-6) which simplifies to 25x + 10y = -30

Now, if we subtract the first equation from the second (or vice versa), we would eliminate the y variable. The option is incorrect, because if you subtract, you would not eliminate a variable. Although, you could solve the equation.

Option 2: Multiply the first equation by 5 and the second equation by 2, then subtract.

If we multiply the first equation by 5 and the second equation by 2, we get:

  • 5 * (-2x + 5y) = 5 * (-15) which simplifies to -10x + 25y = -75
  • 2 * (5x + 2y) = 2 * (-6) which simplifies to 10x + 4y = -12

Now, if we add the equations, the x variable would be eliminated. This is a correct way to solve this equation with the elimination method. This is a correct option.

Step-by-Step Example with the Correct Approach

Let's walk through the correct approach with our system of equations. Remember our equations:

  • -2x + 5y = -15
  • 5x + 2y = -6

  1. Choose a Variable to Eliminate: In this case, we'll eliminate x. We can do this by making the coefficients of x opposites.

  2. Multiply the Equations: Multiply the first equation by 5 and the second equation by 2:

    • 5 * (-2x + 5y = -15) becomes -10x + 25y = -75
    • 2 * (5x + 2y = -6) becomes 10x + 4y = -12

  3. Add the Equations: Now, add the two new equations together:

    • (-10x + 25y) + (10x + 4y) = -75 + (-12)
    • This simplifies to 29y = -87

  4. Solve for y: Divide both sides by 29:

    • y = -87 / 29
    • y = -3

  5. Solve for x: Substitute y = -3 into either of the original equations. Let's use the first equation:

    • -2x + 5(-3) = -15
    • -2x - 15 = -15
    • -2x = 0
    • x = 0

So, the solution to the system of equations is x = 0 and y = -3. Ta-da! We've successfully used the elimination method to solve a system of linear equations. The whole process might seem like a lot of steps at first, but with practice, you'll get super comfortable with it. This is an easy way to solve these types of equations.

Other Strategies and Considerations

Remember that there are many different approaches, and the best one will depend on the specific system of equations you're working with. If the coefficients of one of the variables are already opposites, you can jump straight to adding the equations. Always double-check your work, especially when you're multiplying and dividing. A small mistake can throw off your whole solution. The elimination method is a powerful tool, so now you can solve these equations using it. Make sure you fully understand this. Keep practicing, and soon you'll be a pro at solving systems of linear equations! Let's review what we learned.

Choosing the Right Multipliers

When you're deciding what to multiply the equations by, look for the least common multiple (LCM) of the coefficients you want to eliminate. This helps you choose the smallest numbers to multiply by, making the calculations easier. Another key point when choosing what to eliminate is that your final goal is to eliminate one variable. By following this advice, you can be sure to solve the equation correctly.

Adding or Subtracting? That is the Question

Whether you add or subtract the equations depends on whether the coefficients of the variable you're eliminating have the same sign or opposite signs. If they have opposite signs, add the equations. If they have the same sign, subtract one equation from the other. This will ensure the variable cancels out. This part is important in order to solve the equation.

Checking Your Solution

Always, always check your solution! Plug the x and y values you found back into both original equations to make sure they work. If both equations are true, you know you've got the right answer. This is the final step, make sure you do it.

Wrapping Up

So there you have it! The elimination method is a super effective way to solve systems of linear equations. You can use this method easily to understand, and solve a lot of these equations. Remember the key steps: manipulate the equations to eliminate a variable, solve for the remaining variable, and then substitute that value back into one of the original equations to find the other variable. Keep practicing, and you'll be solving these problems like a math whiz in no time! Remember, practice makes perfect, so work through lots of examples to master this skill. You've got this, guys! Hope you enjoy it!