Solve Using Elimination:$\[ \begin{array}{l} 5x + Y = 18 \\ -3x - Y = -10 \end{array} \\]
Introduction
Solving linear equations is a fundamental concept in mathematics, and one of the most effective methods for solving systems of linear equations is the elimination method. In this article, we will explore the elimination method in detail, using the given system of linear equations as an example.
What is the Elimination Method?
The elimination method is a technique used to solve systems of linear equations by adding or subtracting the equations to eliminate one of the variables. This method is particularly useful when the coefficients of the variables in the two equations are additive inverses of each other.
The Given System of Linear Equations
The given system of linear equations is:
{ \begin{array}{l} 5x + y = 18 \\ -3x - y = -10 \end{array} \}
Step 1: Write Down the Given Equations
The first step in solving the system of linear equations using the elimination method is to write down the given equations.
{ \begin{array}{l} 5x + y = 18 \\ -3x - y = -10 \end{array} \}
Step 2: Multiply the Equations by Necessary Multiples
To eliminate one of the variables, we need to multiply the equations by necessary multiples such that the coefficients of the variable to be eliminated are the same.
Let's multiply the first equation by 1 and the second equation by 1.
{ \begin{array}{l} 5x + y = 18 \\ -3x - y = -10 \end{array} \}
Step 3: Add or Subtract the Equations
Now that we have the equations multiplied by necessary multiples, we can add or subtract the equations to eliminate one of the variables.
Let's add the two equations.
{ \begin{array}{l} (5x + y) + (-3x - y) = 18 + (-10) \\ 2x = 8 \end{array} \}
Step 4: Solve for the Variable
Now that we have eliminated one of the variables, we can solve for the other variable.
Let's solve for x.
{ 2x = 8 \\ x = \frac{8}{2} \\ x = 4 \end{array} \}
Step 5: Substitute the Value of the Variable into One of the Original Equations
Now that we have the value of one of the variables, we can substitute it into one of the original equations to solve for the other variable.
Let's substitute x = 4 into the first equation.
{ 5(4) + y = 18 \\ 20 + y = 18 \\ y = -2 \end{array} \}
Step 6: Write Down the Solution
The solution to the system of linear equations is x = 4 and y = -2.
Conclusion
In this article, we have used the elimination method to solve a system of linear equations. We have shown that 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. The elimination method is a powerful tool for solving systems of linear equations, and it is an essential concept in mathematics.
Frequently Asked Questions
- What is the elimination method? The elimination method is a technique used to solve systems of linear equations by adding or subtracting the equations to eliminate one of the variables.
- How do I use the elimination method to solve a system of linear equations? To use the elimination method, you need to multiply the equations by necessary multiples, add or subtract the equations to eliminate one of the variables, and then solve for the other variable.
- What are the advantages of the elimination method? The elimination method is a powerful tool for solving systems of linear equations because it allows you to eliminate one of the variables and solve for the other variable.
Example Problems
- Solve the system of linear equations using the elimination method:
{ \begin{array}{l} 2x + 3y = 12 \\ x - 2y = -3 \end{array} \}
- Solve the system of linear equations using the elimination method:
{ \begin{array}{l} x + 2y = 6 \\ 3x - 2y = 10 \end{array} \}
Practice Problems
- Solve the system of linear equations using the elimination method:
{ \begin{array}{l} 4x + 2y = 16 \\ 2x - 3y = -5 \end{array} \}
- Solve the system of linear equations using the elimination method:
{ \begin{array}{l} 3x + 2y = 14 \\ x - 4y = -3 \end{array} \}
Final Thoughts
The elimination method is a powerful tool for solving systems of linear equations. By multiplying the equations by necessary multiples and adding or subtracting the equations, you can eliminate one of the variables and solve for the other variable. With practice and patience, you can become proficient in using the elimination method to solve systems of linear equations.
Introduction
The elimination method is a powerful tool for solving systems of linear equations. However, it can be challenging to understand and apply, especially for those who are new to linear algebra. In this article, we will address some of the most frequently asked questions about the elimination method, providing clear and concise answers to help you better understand this technique.
Q: What is the elimination method?
A: The elimination method is a technique used to solve systems of linear equations by adding or subtracting the equations to eliminate one of the variables.
Q: How do I use the elimination method to solve a system of linear equations?
A: To use the elimination method, you need to multiply the equations by necessary multiples, add or subtract the equations to eliminate one of the variables, and then solve for the other variable.
Q: What are the advantages of the elimination method?
A: The elimination method is a powerful tool for solving systems of linear equations because it allows you to eliminate one of the variables and solve for the other variable. It is also a simple and straightforward method that can be applied to a wide range of problems.
Q: What are the disadvantages of the elimination method?
A: One of the main disadvantages of the elimination method is that it can be time-consuming and labor-intensive, especially for large systems of equations. Additionally, it may not be the most efficient method for solving systems of equations with many variables.
Q: When should I use the elimination method?
A: You should use the elimination method when you have a system of linear equations with two variables and the coefficients of the variables are additive inverses of each other.
Q: How do I determine which variable to eliminate?
A: To determine which variable to eliminate, you need to look at the coefficients of the variables in the two equations. If the coefficients are additive inverses of each other, you can eliminate one of the variables.
Q: What if I have a system of linear equations with more than two variables?
A: If you have a system of linear equations with more than two variables, you can use the elimination method to eliminate one of the variables and then use substitution or other methods to solve for the remaining variables.
Q: Can I use the elimination method to solve systems of linear equations with fractions?
A: Yes, you can use the elimination method to solve systems of linear equations with fractions. However, you need to be careful when multiplying and dividing fractions to avoid errors.
Q: What if I have a system of linear equations with decimals?
A: Yes, you can use the elimination method to solve systems of linear equations with decimals. However, you need to be careful when multiplying and dividing decimals to avoid errors.
Q: Can I use the elimination method to solve systems of linear equations with negative numbers?
A: Yes, you can use the elimination method to solve systems of linear equations with negative numbers. However, you need to be careful when multiplying and dividing negative numbers to avoid errors.
Q: What if I have a system of linear equations with no solution?
A: If you have a system of linear equations with no solution, it means that the equations are inconsistent and there is no value of the variables that can satisfy both equations.
Q: What if I have a system of linear equations with infinitely many solutions?
A: If you have a system of linear equations with infinitely many solutions, it means that the equations are dependent and there are many values of the variables that can satisfy both equations.
Conclusion
The elimination method is a powerful tool for solving systems of linear equations. By understanding the advantages and disadvantages of this method, you can apply it effectively to a wide range of problems. Remember to be careful when multiplying and dividing fractions, decimals, and negative numbers, and to check for consistency and dependency in the equations.
Final Thoughts
The elimination method is a fundamental concept in linear algebra, and it is essential to understand this technique to solve systems of linear equations. With practice and patience, you can become proficient in using the elimination method to solve systems of linear equations.