Find The Solution To The System Of Equations Given Below Using Elimination.${ \begin{array}{l} 3x + 2y = -3 \ 9x + 4y = 3 \end{array} }$

Mar 13, 2025 by ADMIN 138 views

**Solving Systems of Equations using Elimination Method**

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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 equations are relatively simple and can be easily manipulated. In this article, we will use the elimination method to solve the system of equations given below.

The System of Equations

The system of equations we will be solving is:

{ \begin{array}{l} 3x + 2y = -3 \\ 9x + 4y = 3 \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 3 and the second equation by 1.

{ \begin{array}{l} 9x + 6y = -9 \\ 9x + 4y = 3 \end{array} \}

Step 2: Subtract the Second Equation from the First Equation

Now that we have the same coefficient for x in both equations, we can subtract the second equation from the first equation to eliminate x.

{ \begin{array}{l} (9x + 6y) - (9x + 4y) = -9 - 3 \\ 2y = -12 \end{array} \}

Step 3: Solve for y

Now that we have eliminated x, we can solve for y by dividing both sides of the equation by 2.

{ \begin{array}{l} y = -12/2 \\ y = -6 \end{array} \}

Step 4: Substitute the Value of 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 = -6 into the first equation.

{ \begin{array}{l} 3x + 2(-6) = -3 \\ 3x - 12 = -3 \end{array} \}

Step 5: Solve for x

Now that we have the value of y substituted into the equation, we can solve for x by adding 12 to both sides of the equation and then dividing both sides by 3.

{ \begin{array}{l} 3x = -3 + 12 \\ 3x = 9 \\ x = 9/3 \\ x = 3 \end{array} \}

Conclusion

In this article, we used the elimination method to solve the system of equations given below.

{ \begin{array}{l} 3x + 2y = -3 \\ 9x + 4y = 3 \end{array} \}

We multiplied the equations by necessary multiples, subtracted the second equation from the first equation, solved for y, substituted the value of y into one of the original equations, and finally solved for x. The solution to the system of equations is x = 3 and y = -6.

Example Problems

Problem 1

Solve the system of equations using the elimination method.

{ \begin{array}{l} 2x + 3y = 7 \\ 4x + 6y = 15 \end{array} \}

Solution

Multiply the first equation by 2 and the second equation by 1.

{ \begin{array}{l} 4x + 6y = 14 \\ 4x + 6y = 15 \end{array} \}

Subtract the first equation from the second equation.

{ \begin{array}{l} (4x + 6y) - (4x + 6y) = 15 - 14 \\ 0 = 1 \end{array} \}

This is a contradiction, so the system of equations has no solution.

Problem 2

Solve the system of equations using the elimination method.

{ \begin{array}{l} x + 2y = 6 \\ 2x + 4y = 12 \end{array} \}

Solution

Multiply the first equation by 2 and the second equation by 1.

{ \begin{array}{l} 2x + 4y = 12 \\ 2x + 4y = 12 \end{array} \}

Subtract the first equation from the second equation.

{ \begin{array}{l} (2x + 4y) - (2x + 4y) = 12 - 12 \\ 0 = 0 \end{array} \}

This is an identity, so the system of equations has infinitely many solutions.

Applications of the Elimination Method

The elimination method has many applications in mathematics and science. Some of the applications include:

Linear Algebra: The elimination method is used to solve systems of linear equations in linear algebra.
Physics: The elimination method is used to solve systems of equations in physics, such as the equations of motion.
Engineering: The elimination method is used to solve systems of equations in engineering, such as the equations of electrical circuits.

Limitations of the Elimination Method

The elimination method has some limitations. Some of the limitations include:

Complex Coefficients: The elimination method is not suitable for systems of equations with complex coefficients.
Non-Linear Equations: The elimination method is not suitable for systems of non-linear equations.
Large Systems: The elimination method can be computationally intensive for large systems of equations.

Conclusion

In this article, we used the elimination method to solve the system of equations given below.

{ \begin{array}{l} 3x + 2y = -3 \\ 9x + 4y = 3 \end{array} \}

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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 the variables in the equations are relatively simple and can be easily manipulated.

Q: How do I choose which variable to eliminate?

A: You can choose which variable to eliminate by looking at the coefficients of the variables in the equations. If the coefficients of one variable are the same in both equations, you can eliminate that variable.

Q: What if the coefficients of the variables are not the same in both equations?

A: If the coefficients of the variables are not the same in both equations, you can multiply the equations by necessary multiples to make the coefficients the same.

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

A: No, the elimination method is not suitable for systems of non-linear equations.

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

A: No, the elimination method is not suitable for systems of equations with complex coefficients.

Q: How do I know if the system of equations has a solution?

A: If the system of equations has a solution, the elimination method will give you a unique solution. If the system of equations has no solution, the elimination method will give you a contradiction. If the system of equations has infinitely many solutions, the elimination method will give you an identity.

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 make the coefficients the same.
Not subtracting the second equation from the first equation to eliminate the variable.
Not solving for the variable after eliminating it.
Not checking if the system of equations has a solution.

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

A: Yes, you can use the elimination method to solve systems of equations with more than two variables. However, it may be more complicated and require more steps.

Q: How do I know if the elimination method is the best method to use for a particular system of equations?

A: You can determine if the elimination method is the best method to use for a particular system of equations by looking at the coefficients of the variables and the complexity of the equations. If the coefficients are relatively simple and the equations are linear, the elimination method may be the best method to use.

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. However, you may need to multiply the equations by necessary multiples to eliminate the fractions.

Q: How do I check if the solution to the system of equations is correct?

A: You can check if the solution to the system of equations is correct by plugging the values of the variables back into the original equations and checking if the equations are satisfied.

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

A: No, the elimination method is not suitable for systems of equations with absolute values.

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

A: No, the elimination method is not suitable for systems of equations with inequalities.

Conclusion

In this article, we answered some frequently asked questions about the 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. It is a useful method for solving systems of equations with relatively simple coefficients and linear equations. However, it may not be suitable for systems of equations with complex coefficients, non-linear equations, or fractions.