Solve Each System By Elimination.${ \begin{array}{l} x - 7y = -2 \ -8x + 7y = 16 \end{array} }$Write Your { X $}$ Answer As Just The Number In The First Blank. In The Second Blank, Write Just Your { Y $}$ Answer. In

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Introduction

Solving systems of equations is a fundamental concept in mathematics, and one of the most effective methods for solving these systems is by elimination. This method involves adding or subtracting equations to eliminate one of the variables, making it easier to solve for the other variable. In this article, we will explore how to solve a system of linear equations using the elimination method.

What is Elimination Method?

The elimination method is a technique used to solve systems of linear equations by adding or subtracting equations to eliminate one of the variables. This method is based on the principle that if two equations are added or subtracted, the resulting equation will have the same coefficients for the variables, but with opposite signs. By eliminating one of the variables, we can solve for the other variable.

Step-by-Step Guide to Solving Systems by Elimination

To solve a system of linear equations using the elimination method, follow these steps:

Step 1: Write Down the Equations

Write down the two equations in the system.

Step 2: Identify the Coefficients

Identify the coefficients of the variables in both equations.

Step 3: Multiply the Equations

Multiply both equations by necessary multiples such that the coefficients of the variable to be eliminated are the same.

Step 4: Add or Subtract the Equations

Add or subtract the equations to eliminate one of the variables.

Step 5: Solve for the Variable

Solve for the variable that is not eliminated.

Step 6: Substitute the Value

Substitute the value of the variable into one of the original equations to solve for the other variable.

Example: Solving a System of Linear Equations

Let's consider the following system of linear equations:

{ \begin{array}{l} x - 7y = -2 \\ -8x + 7y = 16 \end{array} \}

To solve this system using the elimination method, follow the steps outlined above.

Step 1: Write Down the Equations

The two equations in the system are:

x−7y=−2x - 7y = -2

−8x+7y=16-8x + 7y = 16

Step 2: Identify the Coefficients

The coefficients of the variables in both equations are:

xx: 1, -8

yy: -7, 7

Step 3: Multiply the Equations

To eliminate the variable xx, multiply the first equation by 8 and the second equation by 1.

8(x−7y)=8(−2)8(x - 7y) = 8(-2)

−8x+7y=16-8x + 7y = 16

8x−56y=−168x - 56y = -16

−8x+7y=16-8x + 7y = 16

Step 4: Add or Subtract the Equations

Add the two equations to eliminate the variable xx.

(8x−56y)+(−8x+7y)=−16+16(8x - 56y) + (-8x + 7y) = -16 + 16

−49y=−16-49y = -16

Step 5: Solve for the Variable

Solve for the variable yy.

y=−16−49y = \frac{-16}{-49}

y=1649y = \frac{16}{49}

Step 6: Substitute the Value

Substitute the value of yy into one of the original equations to solve for the variable xx.

x−7y=−2x - 7y = -2

x−7(1649)=−2x - 7(\frac{16}{49}) = -2

x−11249=−2x - \frac{112}{49} = -2

x=−2+11249x = -2 + \frac{112}{49}

x=−98+11249x = \frac{-98 + 112}{49}

x=1449x = \frac{14}{49}

x=27x = \frac{2}{7}

Conclusion

In this article, we have explored the elimination method for solving systems of linear equations. This method involves adding or subtracting equations to eliminate one of the variables, making it easier to solve for the other variable. We have also provided a step-by-step guide to solving systems by elimination and applied this method to a system of linear equations. By following these steps and using the elimination method, we can solve systems of linear equations efficiently and accurately.

Final Answer

The final answer is:

x=27x = \frac{2}{7}

Introduction

Solving systems of equations by elimination is a powerful technique used to solve systems of linear equations. However, it can be challenging to understand and apply this method, especially for beginners. In this article, we will address some of the most frequently asked questions about solving systems of equations by elimination.

Q: What is the elimination method?

A: The elimination method is a technique used to solve systems of linear equations by adding or subtracting equations to eliminate one of the variables. This method is based on the principle that if two equations are added or subtracted, the resulting equation will have the same coefficients for the variables, but with opposite signs.

Q: How do I choose which variable to eliminate?

A: To choose which variable to eliminate, look for the coefficients of the variables in both equations. If the coefficients of one variable are the same, but with opposite signs, eliminate that variable. If the coefficients are not the same, multiply both equations by necessary multiples to make the coefficients the same.

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

A: If you have a system of three or more equations, you can use the elimination method to solve for two variables, and then use substitution or elimination to solve for the remaining variables.

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

A: Yes, you can use the elimination method to solve systems with fractions. However, you may need to multiply both equations by the least common multiple (LCM) of the denominators to eliminate the fractions.

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

A: To check if you have made a mistake in the elimination method, plug the values of the variables back into the original equations to see if they are true. If the values do not satisfy the original equations, you may have made a mistake.

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

A: Yes, you can use the elimination method to solve systems with decimals. However, you may need to multiply both equations by a power of 10 to eliminate the decimals.

Q: How do I choose which equation to add or subtract?

A: To choose which equation to add or subtract, look for the coefficients of the variables in both equations. If the coefficients of one variable are the same, but with opposite signs, add or subtract the equations to eliminate that variable.

Q: Can I use the elimination method to solve systems with absolute values?

A: Yes, you can use the elimination method to solve systems with absolute values. However, you may need to consider multiple cases to account for the absolute values.

Q: How do I know if I have found the correct solution?

A: To check if you have found the correct solution, plug the values of the variables back into the original equations to see if they are true. If the values do not satisfy the original equations, you may have made a mistake.

Conclusion

In this article, we have addressed some of the most frequently asked questions about solving systems of equations by elimination. By understanding the elimination method and its applications, you can solve systems of linear equations efficiently and accurately. Remember to always check your work by plugging the values of the variables back into the original equations to ensure that you have found the correct solution.

Final Answer

The final answer is:

  • The elimination method is a technique used to solve systems of linear equations by adding or subtracting equations to eliminate one of the variables.
  • To choose which variable to eliminate, look for the coefficients of the variables in both equations.
  • You can use the elimination method to solve systems with fractions, decimals, and absolute values.
  • To check if you have found the correct solution, plug the values of the variables back into the original equations to see if they are true.