Solve The System Of Equations Using Elimination.${ \begin{array}{l} 7x + 2y = -13 \ -7x + Y = 25 \end{array} }$
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 linear equations. 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 learn how to solve a system of 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 concept of adding or subtracting equations to eliminate one of the variables, making it easier to solve for the other variable. The elimination method is a powerful tool for solving systems of linear equations and is widely used in mathematics and science.
How to Solve a System of Equations Using Elimination
To solve a system of equations using elimination, we need to follow these steps:
- Write down the equations: Write down the two equations that make up the system of equations.
- Identify the coefficients: Identify the coefficients of the variables in each equation.
- Multiply the equations: Multiply one or both of the equations by a number that will make the coefficients of one of the variables the same in both equations.
- Add or subtract the equations: Add or subtract the equations to eliminate one of the variables.
- Solve for the other variable: Solve for the other variable using the resulting equation.
- Check the solution: Check the solution by plugging it back into both original equations.
Step-by-Step Solution
Let's use the following system of equations as an example:
{ \begin{array}{l} 7x + 2y = -13 \\ -7x + y = 25 \end{array} \}
To solve this system of equations using elimination, we need to follow the steps outlined above.
Step 1: Write down the equations
The two equations that make up the system of equations are:
{ \begin{array}{l} 7x + 2y = -13 \\ -7x + y = 25 \end{array} \}
Step 2: Identify the coefficients
The coefficients of the variables in each equation are:
- Equation 1: 7x and 2y
- Equation 2: -7x and y
Step 3: Multiply the equations
To make the coefficients of one of the variables the same in both equations, we can multiply Equation 1 by 1 and Equation 2 by 2.
{ \begin{array}{l} 7x + 2y = -13 \\ -14x + 2y = 50 \end{array} \}
Step 4: Add or subtract the equations
Now that we have the coefficients of one of the variables the same in both equations, we can add or subtract the equations to eliminate one of the variables. Let's subtract Equation 1 from Equation 2.
{ \begin{array}{l} 7x + 2y = -13 \\ -21x = 63 \end{array} \}
Step 5: Solve for the other variable
Now that we have eliminated one of the variables, we can solve for the other variable using the resulting equation.
{ -21x = 63 \}
Dividing both sides of the equation by -21, we get:
{ x = -3 \}
Step 6: Check the solution
To check the solution, we need to plug it back into both original equations.
{ \begin{array}{l} 7x + 2y = -13 \\ -7x + y = 25 \end{array} \}
Plugging x = -3 into both equations, we get:
{ \begin{array}{l} 7(-3) + 2y = -13 \\ -7(-3) + y = 25 \end{array} \}
Simplifying both equations, we get:
{ \begin{array}{l} -21 + 2y = -13 \\ 21 + y = 25 \end{array} \}
Adding 21 to both sides of the first equation and subtracting 21 from both sides of the second equation, we get:
{ \begin{array}{l} 2y = 8 \\ y = 4 \end{array} \}
Therefore, the solution to the system of equations is x = -3 and y = 4.
Conclusion
In this article, we learned how to solve a system of equations using the elimination method. We used the following steps to solve the system of equations:
- Write down the equations
- Identify the coefficients
- Multiply the equations
- Add or subtract the equations
- Solve for the other variable
- Check the solution
We also used a step-by-step example to illustrate how to solve a system of equations using the elimination method. By following these steps, we can solve systems of linear equations and find the values of the variables.
Frequently Asked Questions
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: How do I know which variable to eliminate?
A: To determine which variable to eliminate, you need to look at the coefficients of the variables in each equation. If the coefficients of one of the variables are the same in both equations, you can eliminate that variable.
Q: Can I use the elimination method to solve systems of nonlinear equations?
A: No, the elimination method is only used to solve systems of linear equations. Nonlinear equations require different methods, such as substitution or graphing.
Q: What if I get stuck during the elimination process?
A: If you get stuck during the elimination process, you can try multiplying one or both of the equations by a number that will make the coefficients of one of the variables the same in both equations.
References
- [1] "Algebra and Trigonometry" by Michael Sullivan
- [2] "College Algebra" by James Stewart
- [3] "Linear Algebra and Its Applications" by Gilbert Strang
Additional Resources
- Khan Academy: Systems of Equations
- Mathway: Systems of Equations
- Wolfram Alpha: Systems of Equations
Solve the System of Equations Using Elimination: Q&A =====================================================
Introduction
In our previous article, we learned how to solve a system of equations using the elimination method. In this article, we will answer some frequently asked questions about the elimination method and provide additional resources for further learning.
Q&A
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: How do I know which variable to eliminate?
A: To determine which variable to eliminate, you need to look at the coefficients of the variables in each equation. If the coefficients of one of the variables are the same in both equations, you can eliminate that variable.
Q: Can I use the elimination method to solve systems of nonlinear equations?
A: No, the elimination method is only used to solve systems of linear equations. Nonlinear equations require different methods, such as substitution or graphing.
Q: What if I get stuck during the elimination process?
A: If you get stuck during the elimination process, you can try multiplying one or both of the equations by a number that will make the coefficients of one of the variables the same in both equations.
Q: How do I know if the elimination method is the best method to use?
A: The elimination method is a good method to use when the coefficients of the variables are the same in both equations. However, if the coefficients are not the same, you may need to use a different method, such as substitution or graphing.
Q: Can I use the elimination method to solve systems of equations with more than two variables?
A: Yes, the elimination method can be used to solve systems of equations with more than two variables. However, it may be more complicated and require more steps.
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 checking the solution by plugging it back into both original equations
- Not following the correct steps to eliminate the variable
- Not using the correct method for the type of equation
Q: How do I check the solution?
A: To check the solution, you need to plug it back into both original equations and make sure it satisfies both equations.
Q: What if I get a false solution?
A: If you get a false solution, it means that the solution does not satisfy both equations. You need to go back and check your work to see where you made a mistake.
Additional Resources
- Khan Academy: Systems of Equations
- Mathway: Systems of Equations
- Wolfram Alpha: Systems of Equations
- Algebra.com: Systems of Equations
- Purplemath: Systems of Equations
Practice Problems
- Solve the system of equations using the elimination method:
{ \begin{array}{l} 2x + 3y = 7 \\ x - 2y = -3 \end{array} \}
- Solve the system of equations using the elimination method:
{ \begin{array}{l} x + 2y = 4 \\ 3x - 2y = 5 \end{array} \}
- Solve the system of equations using the elimination method:
{ \begin{array}{l} x + y = 3 \\ 2x - 3y = -6 \end{array} \}
Conclusion
In this article, we answered some frequently asked questions about the elimination method and provided additional resources for further learning. We also provided practice problems for you to try. Remember to always check your solution by plugging it back into both original equations and to avoid common mistakes when using the elimination method.
References
- [1] "Algebra and Trigonometry" by Michael Sullivan
- [2] "College Algebra" by James Stewart
- [3] "Linear Algebra and Its Applications" by Gilbert Strang
Glossary
- Elimination method: A technique used to solve systems of linear equations by adding or subtracting equations to eliminate one of the variables.
- Coefficient: A number that is multiplied by a variable in an equation.
- Variable: A letter or symbol that represents a value in an equation.
- System of equations: A set of two or more equations that are solved simultaneously to find the values of the variables.