The elimination method is a technique used to solve systems of linear equations. The basic idea is to eliminate one variable at a time, allowing us to solve for the remaining variable. This technique is especially useful when dealing with two variables because it simplifies the calculations.
Let’s consider an example to illustrate the elimination method:
Imagine we have the following system of equations:
- Equation 1: 2x + 3y = 12
- Equation 2: 4x – y = 5
Step 1: Align the equations
2x + 3y = 12 (1) 4x - y = 5 (2)
Step 2: Multiply the equations if necessary to make the coefficients of one variable the same. In this case, we can multiply Equation 2 by 3 to align the coefficients of y:
2x + 3y = 12 (1) 12x - 3y = 15 (2) * 3
Step 3: Add the equations to eliminate y:
(2x + 3y) + (12x - 3y) = 12 + 15
This results in:
14x = 27
Step 4: Solve for x:
x = 27 / 14 = 1.93 (approximately)
Step 5: Substitute x back into one of the original equations to find y. We’ll use Equation 1:
2(1.93) + 3y = 12
This simplifies to:
3.86 + 3y = 12
Step 6: Isolate y:
3y = 12 - 3.86 = 8.14
y = 8.14 / 3 = 2.71 (approximately)
Thus, the solution to the system of equations is approximately:
x ≈ 1.93, y ≈ 2.71
In summary, the elimination method allows us to systematically eliminate variables to simplify and solve systems of equations effectively. By following these steps carefully, you can arrive at the correct solutions.