How To Do System Of Equations
System of equations can be solved using various methods such as graphing, substitution, and elimination. In this section, we will focus on the elimination method and learn how to solve a system of equations using this approach. The elimination method is based on the Addition Property of Equality, which states that when you add equal quantities to both sides of an equation, the results are equal.
Solve a System of Equations by Elimination
The first step in solving a system of equations by elimination is to write both equations in standard form. Then, we decide which variable will be easiest to eliminate. We want to have the coefficients of one variable be opposites, so that we can add the equations together and eliminate that variable. If the coefficients are not opposites, we can multiply one or both equations by a constant to make the coefficients opposites. Once we have an equation with just one variable, we solve it and substitute that value into one of the original equations to solve for the remaining variable. Finally, we check our answer to ensure it is a solution to both of the original equations.
Example Problems
Let’s solve a few example problems using the elimination method:
Problem 1
Solve the system by elimination. (left{begin{array}{l}{3 x+y=5} \ {2 x-3 y=7}end{array}right.)
Answer: (2,−1)
In this problem, we can make the coefficients of the variable y opposites by multiplying the second equation by 3. After adding the equations, we get a single equation with one variable, which can be easily solved.
Problem 2
Solve the system by elimination. (left{begin{array}{l}{4 x+y=-5} \ {-2 x-2 y=-2}end{array}right.)
Answer: (−2,3)
In this problem, we can directly add the equations to eliminate one variable and then solve for the remaining variable.
Problem 3
Solve the system by elimination. (left{begin{array}{l}{x+y=3} \ {-2 x-y=-1}end{array}right.)
Answer: (−2,5)
In this problem, we can make the coefficients of the variable y opposites by multiplying the second equation by 2. After adding the equations, we get a single equation with one variable, which can be easily solved.
FAQs
Q: What is the Elimination Method?
The Elimination Method is a technique used to solve a system of linear equations by adding the equations together to eliminate one of the variables.
Q: When should the Elimination Method be used?
The Elimination Method should be used when the coefficients of one variable in the equations can be made opposites by multiplying one or both equations by a constant.
Q: What should be done after obtaining the solution using the Elimination Method?
After obtaining the solution, it is important to substitute the values back into the original equations to ensure that they satisfy both equations.
Q: Can the Elimination Method be used for systems with fractions?
Yes, the Elimination Method can be used for systems with fractions. The first step in such cases is to clear the fractions by multiplying each equation by its least common denominator (LCD).
Q: How can I check if the solution obtained is correct?
To check if the solution is correct, substitute the values back into the original equations and verify that they satisfy both equations.