The System Of Equations Can Be Solved Using Linear Combination To Eliminate One Of The Variables.$\[ \begin{array}{l} 2x - Y = -4 \rightarrow 10x - 5y = -20 \\ 3x + 5y = 59 \rightarrow \begin{array}{c} 3x + 5y = 59 \\ 13x =
**The System of Equations: A Comprehensive Guide to Solving Using Linear Combination**
What is a System of Equations?
A system of equations is a set of two or more equations that contain multiple variables. These equations are related to each other and must be solved simultaneously to find the values of the variables. In this article, we will focus on solving systems of equations using linear combination, a method that allows us to eliminate one of the variables by adding or subtracting the equations.
What is Linear Combination?
Linear combination is a method used to solve systems of equations by adding or subtracting the equations to eliminate one of the variables. This method involves multiplying one or more of the equations by a constant and then adding or subtracting the resulting equations to eliminate one of the variables.
How to Solve a System of Equations Using Linear Combination
To solve a system of equations using linear combination, follow these steps:
- Write down the system of equations: Start by writing down the system of equations that you want to solve.
- Identify the variables: Identify the variables in the system of equations.
- Choose a method: Choose a method to solve the system of equations, such as linear combination.
- Multiply one or more equations by a constant: Multiply one or more of the equations by a constant to make the coefficients of one of the variables the same.
- Add or subtract the resulting equations: Add or subtract the resulting equations to eliminate one of the variables.
- Solve for the remaining variable: Solve for the remaining variable using the resulting equation.
- Check the solution: Check the solution by plugging it back into the original equations.
Example: Solving a System of Equations Using Linear Combination
Let's consider the following system of equations:
2x - y = -4 3x + 5y = 59
To solve this system of equations using linear combination, we can multiply the first equation by 5 and the second equation by 1.
5(2x - y) = 5(-4) 3x + 5y = 59
This gives us:
10x - 5y = -20 3x + 5y = 59
Now, we can add the two equations to eliminate the variable y.
(10x - 5y) + (3x + 5y) = -20 + 59 13x = 39
Now, we can solve for x by dividing both sides by 13.
x = 39/13 x = 3
Now that we have found the value of x, we can substitute it back into one of the original equations to find the value of y.
2x - y = -4 2(3) - y = -4 6 - y = -4 -y = -10 y = 10
Therefore, the solution to the system of equations is x = 3 and y = 10.
Frequently Asked Questions
Q: What is the difference between linear combination and substitution?
A: Linear combination and substitution are two different methods used to solve systems of equations. Linear combination involves adding or subtracting the equations to eliminate one of the variables, while substitution involves solving one of the equations for one of the variables and then substituting it into the other equation.
Q: How do I know which method to use?
A: The choice of method depends on the system of equations. If the coefficients of one of the variables are the same in both equations, linear combination is a good choice. If the coefficients are different, substitution may be a better option.
Q: Can I use linear combination to solve a system of three or more equations?
A: Yes, linear combination can be used to solve a system of three or more equations. However, it may be more complicated and require more steps.
Q: What if I get stuck during the process?
A: If you get stuck during the process, try to simplify the equations by multiplying or dividing both sides by a constant. You can also try to use a different method, such as substitution.
Conclusion
Solving systems of equations using linear combination is a powerful tool that can be used to find the values of multiple variables. By following the steps outlined in this article, you can use linear combination to solve systems of equations and find the values of the variables. Remember to check your solution by plugging it back into the original equations to ensure that it is correct.