Solve The Following System Using The Elimination Method.${ \begin{array}{l} 6x - 5y = 22 \ 2x + 6y = -8 \end{array} }$

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Introduction

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 particularly useful when the coefficients of the variables in the two equations are multiples of each other. In this article, we will use the elimination method to solve the following system of linear equations:

{ \begin{array}{l} 6x - 5y = 22 \\ 2x + 6y = -8 \end{array} \}

Step 1: Multiply the Equations by Necessary Multiples

To eliminate one of the variables, we need to make the coefficients of either x or y the same in both equations. We can do this by multiplying the equations by necessary multiples.

Let's multiply the first equation by 2 and the second equation by 6:

{ \begin{array}{l} 12x - 10y = 44 \\ 12x + 36y = -48 \end{array} \}

Step 2: Add or Subtract the Equations

Now that we have the coefficients of x the same in both equations, we can add or subtract the equations to eliminate x. Let's add the two equations:

{ (12x - 10y) + (12x + 36y) = 44 + (-48) \}

Simplifying the equation, we get:

{ 24x + 26y = -4 \}

However, we made a mistake in our previous step. We should have subtracted the second equation from the first equation to eliminate x. Let's correct this:

{ (12x - 10y) - (12x + 36y) = 44 - (-48) \}

Simplifying the equation, we get:

{ -46y = 92 \}

Step 3: Solve for y

Now that we have the equation -46y = 92, we can solve for y by dividing both sides by -46:

{ y = -\frac{92}{46} \}

Simplifying the equation, we get:

{ y = -2 \}

Step 4: 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 substitute y into the first equation:

{ 6x - 5(-2) = 22 \}

Simplifying the equation, we get:

{ 6x + 10 = 22 \}

Subtracting 10 from both sides, we get:

{ 6x = 12 \}

Dividing both sides by 6, we get:

{ x = 2 \}

Conclusion

In this article, we used the elimination method to solve the following system of linear equations:

{ \begin{array}{l} 6x - 5y = 22 \\ 2x + 6y = -8 \end{array} \}

By multiplying the equations by necessary multiples, adding or subtracting the equations, solving for y, and substituting y into one of the original equations, we found the values of x and y to be x = 2 and y = -2.

Example Use Cases

The elimination method can be used to solve a wide range of systems of linear equations, including:

  • Business: A company has two factories that produce two different products. The cost of producing x units of product A and y units of product B is given by the equations 6x - 5y = 22 and 2x + 6y = -8. Using the elimination method, we can find the optimal production levels for the two factories.
  • Science: A scientist is studying the relationship between the concentration of two chemicals in a solution. The concentration of chemical A is given by the equation 6x - 5y = 22, and the concentration of chemical B is given by the equation 2x + 6y = -8. Using the elimination method, we can find the concentrations of the two chemicals.
  • Engineering: An engineer is designing a system that involves two variables. The system is described by the equations 6x - 5y = 22 and 2x + 6y = -8. Using the elimination method, we can find the values of the two variables.

Advantages and Disadvantages

The elimination method has several advantages, including:

  • Easy to understand: The elimination method is a simple and intuitive technique that is easy to understand.
  • Fast: The elimination method is a fast technique that can be used to solve systems of linear equations quickly.
  • Accurate: The elimination method is an accurate technique that can be used to find the exact values of the variables.

However, the elimination method also has some disadvantages, including:

  • Limited applicability: The elimination method is only applicable to systems of linear equations that can be solved using this technique.
  • Requires careful manipulation: The elimination method requires careful manipulation of the equations to eliminate one of the variables.
  • May not be suitable for all systems: The elimination method may not be suitable for all systems of linear equations, particularly those that involve complex or nonlinear relationships.
    Frequently Asked Questions (FAQs) about the Elimination Method ====================================================================

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.

Q: When can I use the elimination method?

A: You can use the elimination method when the coefficients of either x or y are the same in both equations, or when the coefficients are multiples of each other.

Q: How do I multiply the equations by necessary multiples?

A: To multiply the equations by necessary multiples, you need to find the least common multiple (LCM) of the coefficients of either x or y. Then, multiply both equations by the LCM.

Q: What is the difference between adding and subtracting the equations?

A: Adding the equations eliminates one of the variables, while subtracting the equations eliminates the other variable.

Q: How do I solve for y?

A: To solve for y, you need to isolate y on one side of the equation. This can be done by adding or subtracting the same value to both sides of the equation.

Q: How do I substitute y into one of the original equations?

A: To substitute y into one of the original equations, you need to replace y with its value in the equation.

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
  • Not adding or subtracting the equations correctly
  • Not solving for y correctly
  • Not substituting y into one of the original equations correctly

Q: Can I use the elimination method to solve systems of linear equations with more than two variables?

A: No, the elimination method is only applicable to systems of linear equations with two variables.

Q: Can I use the elimination method to solve systems of linear equations with complex or nonlinear relationships?

A: No, the elimination method is only applicable to systems of linear equations with linear relationships.

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

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

  • Business: The elimination method can be used to solve systems of linear equations that describe the cost of producing different products.
  • Science: The elimination method can be used to solve systems of linear equations that describe the relationship between different variables in a scientific experiment.
  • Engineering: The elimination method can be used to solve systems of linear equations that describe the behavior of different systems in engineering.

Q: How do I choose between the elimination method and other methods, such as substitution or graphing?

A: You should choose the elimination method when:

  • The coefficients of either x or y are the same in both equations
  • The coefficients are multiples of each other
  • You need to find the values of both variables

You should choose substitution or graphing when:

  • The coefficients of either x or y are not the same in both equations
  • The coefficients are not multiples of each other
  • You only need to find the value of one variable

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

A: Yes, you can use the elimination method to solve systems of linear equations with fractions or decimals. However, you need to be careful when multiplying the equations by necessary multiples, as this can introduce fractions or decimals into the equations.