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If A, B is a Solution of the System of Equations: A Comprehensive Guide
Solving systems of equations is a fundamental concept in algebra. It involves finding the values of variables that satisfy all equations in the system simultaneously. This comprehensive guide will walk you through various methods to solve such systems and explain how to verify if a given pair (A, B) is indeed a solution.
Understanding Systems of Equations
A system of equations is a collection of two or more equations with the same variables. The goal is to find values for the variables that make all equations true. For example:
- Equation 1: 2x + y = 7
- Equation 2: x - y = 2
Here, we have a system of two linear equations with two variables, x and y. A solution to this system is a pair of values (x, y) that satisfies both equations.
Methods for Solving Systems of Equations
Several methods can effectively solve systems of equations. The most common ones are:
1. Substitution Method
This method involves solving one equation for one variable and substituting the expression into the other equation. Let's apply it to our example:
- Solve for one variable: From Equation 2, we can easily solve for x: x = y + 2
- Substitute: Substitute this expression for x (y + 2) into Equation 1: 2(y + 2) + y = 7
- Solve for the remaining variable: Simplify and solve for y: 2y + 4 + y = 7 => 3y = 3 => y = 1
- Substitute back: Substitute the value of y (1) back into either Equation 1 or 2 to find x. Using Equation 2: x - 1 = 2 => x = 3
- Solution: The solution is (x, y) = (3, 1)
2. Elimination Method
The elimination method, also known as the addition method, involves manipulating the equations to eliminate one variable by adding or subtracting them. For our example:
- Add the equations: Notice that the 'y' terms have opposite signs. Adding Equation 1 and Equation 2 directly eliminates y: (2x + y) + (x - y) = 7 + 2 => 3x = 9 => x = 3
- Substitute: Substitute x = 3 into either Equation 1 or 2 to solve for y. Using Equation 2: 3 - y = 2 => y = 1
- Solution: The solution remains (x, y) = (3, 1)
3. Graphical Method
This method involves graphing each equation on a coordinate plane. The point where the lines intersect represents the solution.
Verifying a Solution (A, B)
Once you've found a potential solution (A, B), it's crucial to verify it by substituting the values into all equations in the system. If both equations are true after substitution, then (A, B) is indeed a solution.
For our example, substituting x = 3 and y = 1 into both equations:
- Equation 1: 2(3) + 1 = 7 (True)
- Equation 2: 3 - 1 = 2 (True)
Since both equations hold true, (3, 1) is confirmed as the solution.
Handling Different Types of Systems
- Consistent and Independent Systems: These systems have exactly one unique solution (like our example).
- Consistent and Dependent Systems: These systems have infinitely many solutions. The equations represent the same line.
- Inconsistent Systems: These systems have no solution. The lines are parallel and never intersect.
Conclusion
Solving systems of equations is a powerful tool with applications across various fields. Mastering the substitution, elimination, and graphical methods empowers you to tackle diverse problems efficiently. Always remember to verify your solution to ensure accuracy. By understanding these concepts and practicing regularly, you'll gain confidence and proficiency in solving even complex systems of equations.
