Solve The Following System Of Equations Using The Elimination Method:${ \begin{array}{l} 8x + Y = -16 \ -3x + Y = -5 \end{array} }$

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Introduction

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. The elimination method is one of the most common techniques used to solve systems of equations. This method involves adding or subtracting the equations to eliminate one of the variables, making it easier to solve for the other variable. In this article, we will use the elimination method to solve a system of two linear equations.

The System of Equations

The system of equations we will be solving is:

{ \begin{array}{l} 8x + y = -16 \\ -3x + y = -5 \end{array} \}

Step 1: Write Down the Equations

The first step in solving the system of equations is to write down the equations. In this case, we have two equations:

  1. 8x+y=−168x + y = -16
  2. −3x+y=−5-3x + y = -5

Step 2: Multiply the Equations by Necessary Multiples

To eliminate one of the variables, we need to multiply the equations by necessary multiples. We can multiply the first equation by 3 and the second equation by 8 to make the coefficients of y's in both equations equal.

  1. 24x+3y=−4824x + 3y = -48
  2. −24x+8y=−40-24x + 8y = -40

Step 3: Add the Equations

Now that we have the equations with the same coefficients for y, we can add them to eliminate the y variable.

24x+3y−24x+8y=−48−4024x + 3y -24x + 8y = -48 -40

This simplifies to:

11y=−8811y = -88

Step 4: Solve for y

Now that we have the equation with only one variable, we can solve for y.

y=−8811y = \frac{-88}{11}

y=−8y = -8

Step 5: Substitute y into One of the Original Equations

Now that we have the value of y, we can substitute it into one of the original equations to solve for x.

Let's use the first equation:

8x+y=−168x + y = -16

Substituting y = -8, we get:

8x−8=−168x - 8 = -16

Step 6: Solve for x

Now that we have the equation with only one variable, we can solve for x.

8x=−16+88x = -16 + 8

8x=−88x = -8

x=−88x = \frac{-8}{8}

x=−1x = -1

Conclusion

In this article, we used the elimination method to solve a system of two linear equations. We multiplied the equations by necessary multiples, added the equations to eliminate the y variable, solved for y, and then substituted y into one of the original equations to solve for x. The final solution is x = -1 and y = -8.

Why Use the Elimination Method?

The elimination method is a powerful technique for solving systems of equations. It is particularly useful when the coefficients of the variables are large or when the variables are not easily isolated. By multiplying the equations by necessary multiples and adding or subtracting the equations, we can eliminate one of the variables and solve for the other variable.

Real-World Applications

The elimination method has many real-world applications. For example, in physics, we can use the elimination method to solve systems of equations that describe the motion of objects. In economics, we can use the elimination method to solve systems of equations that describe the behavior of markets. In engineering, we can use the elimination method to solve systems of equations that describe the behavior of complex systems.

Tips and Tricks

Here are some tips and tricks for using the elimination method:

  • Make sure to multiply the equations by necessary multiples to make the coefficients of the variables equal.
  • Add or subtract the equations to eliminate one of the variables.
  • Solve for the variable that is easiest to isolate.
  • Check your solution by plugging it back into the original equations.

Conclusion

Q: What is the elimination method?

A: The elimination method is a technique used to solve systems of equations by adding or subtracting the equations to eliminate one of the variables.

Q: When should I use the elimination method?

A: You should use the elimination method when the coefficients of the variables are large or when the variables are not easily isolated.

Q: How do I know which variable to eliminate?

A: You should eliminate the variable that is easiest to isolate. In other words, you should eliminate the variable that has the smallest coefficient.

Q: What if I have a system of equations with three or more variables?

A: In that case, you can use the elimination method to eliminate two variables at a time, or you can use other methods such as substitution or matrices.

Q: Can I use the elimination method to solve non-linear systems of equations?

A: No, the elimination method is only used to solve linear systems of equations. If you have a non-linear system of equations, you will need to use other methods such as substitution or matrices.

Q: How do I know if I have made a mistake in the elimination method?

A: You can check your solution by plugging it back into the original equations. If the solution does not satisfy the original equations, then you have made a mistake.

Q: Can I use the elimination method to solve systems of equations with fractions?

A: Yes, you can use the elimination method to solve systems of equations with fractions. Just make sure to multiply the equations by necessary multiples to eliminate the fractions.

Q: What are some common mistakes to avoid when using the elimination method?

A: Some common mistakes to avoid when using the elimination method include:

  • Not multiplying the equations by necessary multiples to eliminate the fractions.
  • Not adding or subtracting the equations correctly.
  • Not solving for the variable that is easiest to isolate.
  • Not checking the solution by plugging it back into the original equations.

Q: Can I use the elimination method to solve systems of equations with decimals?

A: Yes, you can use the elimination method to solve systems of equations with decimals. Just make sure to multiply the equations by necessary multiples to eliminate the decimals.

Q: What are some real-world applications of the elimination method?

A: Some real-world applications of the elimination method include:

  • Solving systems of equations that describe the motion of objects in physics.
  • Solving systems of equations that describe the behavior of markets in economics.
  • Solving systems of equations that describe the behavior of complex systems in engineering.

Q: Can I use the elimination method to solve systems of equations with negative numbers?

A: Yes, you can use the elimination method to solve systems of equations with negative numbers. Just make sure to multiply the equations by necessary multiples to eliminate the negative numbers.

Q: What are some tips and tricks for using the elimination method?

A: Some tips and tricks for using the elimination method include:

  • Make sure to multiply the equations by necessary multiples to make the coefficients of the variables equal.
  • Add or subtract the equations to eliminate one of the variables.
  • Solve for the variable that is easiest to isolate.
  • Check your solution by plugging it back into the original equations.

Conclusion

In conclusion, the elimination method is a powerful technique for solving systems of equations. By adding or subtracting the equations to eliminate one of the variables, we can solve for the other variable. The elimination method has many real-world applications and is a useful tool for solving systems of equations.