Use The Elimination Method To Solve The System:${ \begin{align*} 2s + T &= -5 \ -2s - 3t &= -9 \end{align*} }$Find The Values Of { (s, T)$}$.
Introduction
The elimination method is a powerful technique used to solve systems of linear equations. It 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 use the elimination method to solve the system of linear equations:
{ \begin{align*} 2s + t &= -5 \\ -2s - 3t &= -9 \end{align*} \}
Understanding the System of Linear Equations
The given system of linear equations consists of two equations with two variables, s and t. The first equation is , and the second equation is . Our goal is to find the values of s and t that satisfy both equations.
Step 1: Multiply the Equations by Necessary Multiples
To eliminate one of the variables, we need to make the coefficients of either s or t the same in both equations, but with opposite signs. We can do this by multiplying the first equation by 3 and the second equation by 1.
{ \begin{align*} (3)(2s + t) &= (3)(-5) \\ (1)(-2s - 3t) &= (1)(-9) \end{align*} \}
This gives us:
{ \begin{align*} 6s + 3t &= -15 \\ -2s - 3t &= -9 \end{align*} \}
Step 2: Add the Two Equations
Now that we have the coefficients of t as 3 and -3, we can add the two equations to eliminate the variable t.
{ \begin{align*} (6s + 3t) + (-2s - 3t) &= -15 + (-9) \\ 6s - 2s &= -24 \\ 4s &= -24 \end{align*} \}
Step 3: Solve for s
Now that we have the equation , we can solve for s by dividing both sides by 4.
{ \begin{align*} \frac{4s}{4} &= \frac{-24}{4} \\ s &= -6 \end{align*} \}
Step 4: Substitute s into One of the Original Equations
Now that we have the value of s, we can substitute it into one of the original equations to solve for t. We will use the first equation: .
{ \begin{align*} 2(-6) + t &= -5 \\ -12 + t &= -5 \\ t &= 7 \end{align*} \}
Conclusion
Using the elimination method, we have solved the system of linear equations:
{ \begin{align*} 2s + t &= -5 \\ -2s - 3t &= -9 \end{align*} \}
The values of s and t that satisfy both equations are s = -6 and t = 7.
Advantages of the Elimination Method
The elimination method has several advantages, including:
- It is a simple and straightforward method to use.
- It can be used to solve systems of linear equations with any number of variables.
- It can be used to solve systems of linear equations with any type of coefficients (e.g. integers, fractions, decimals).
Disadvantages of the Elimination Method
The elimination method also has some disadvantages, including:
- It can be time-consuming to use, especially for large systems of linear equations.
- It can be difficult to use if the coefficients of the variables are not easily factorable.
- It can be difficult to use if the system of linear equations has no solution.
Real-World Applications of the Elimination Method
The elimination method has many real-world applications, including:
- Physics and Engineering: The elimination method is used to solve systems of linear equations that arise in physics and engineering, such as the motion of objects under the influence of gravity.
- Computer Science: The elimination method is used to solve systems of linear equations that arise in computer science, such as the solution of linear programming problems.
- Economics: The elimination method is used to solve systems of linear equations that arise in economics, such as the solution of linear programming problems.
Conclusion
In conclusion, the elimination method is a powerful technique used to solve systems of linear equations. It involves adding or subtracting equations to eliminate one of the variables, making it easier to solve for the other variable. The elimination method has several advantages, including its simplicity and ease of use. However, it also has some disadvantages, including its potential for time-consuming calculations and difficulty in using it for large systems of linear equations. Despite these disadvantages, the elimination method remains a widely used and effective technique for solving systems of linear equations.
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, making it easier to solve for the other variable.
Q: How do I know which variable to eliminate first?
A: To determine which variable to eliminate first, you need to look at the coefficients of the variables in the two equations. If the coefficients of one variable are the same in both equations, but with opposite signs, you can eliminate that variable by adding or subtracting the two equations.
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 one or both of the equations by a constant to make the coefficients the same. This is called "multiplying by a necessary multiple".
Q: How do I know which equation to multiply by a necessary multiple?
A: To determine which equation to multiply by a necessary multiple, you need to look at the coefficients of the variable you want to eliminate. You want to make the coefficients of that variable the same in both equations, but with opposite signs.
Q: What if I multiply one equation by a necessary multiple and it doesn't work?
A: If you multiply one equation by a necessary multiple and it doesn't work, you can try multiplying the other equation by a different necessary multiple. You can also try multiplying both equations by different necessary multiples.
Q: How do I know when to stop multiplying by necessary multiples?
A: You know when to stop multiplying by necessary multiples when the coefficients of the variable you want to eliminate are the same in both equations, but with opposite signs.
Q: What if I get a contradiction when I add or subtract the two equations?
A: If you get a contradiction when you add or subtract the two equations, it means that the system of linear equations has no solution.
Q: What if I get a true statement when I add or subtract the two equations?
A: If you get a true statement when you add or subtract the two equations, it means that the system of linear equations has infinitely many solutions.
Q: Can I use the elimination method to solve systems of linear equations with more than two variables?
A: Yes, you can use the elimination method to solve systems of linear equations with more than two variables. You just need to eliminate one variable at a time, using the same steps as before.
Q: Is the elimination method always the best method to use?
A: No, the elimination method is not always the best method to use. Sometimes, other methods such as substitution or graphing may be more efficient or easier to use.
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. You just need to follow the same steps as before, using the fractions or decimals as coefficients.
Q: Can I use the elimination method to solve systems of linear equations with complex numbers?
A: Yes, you can use the elimination method to solve systems of linear equations with complex numbers. You just need to follow the same steps as before, using the complex numbers as coefficients.
Conclusion
In conclusion, the elimination method is a powerful technique used to solve systems of linear equations. It involves adding or subtracting equations to eliminate one of the variables, making it easier to solve for the other variable. By following the steps outlined in this article, you can use the elimination method to solve systems of linear equations with any number of variables, including fractions, decimals, and complex numbers.