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Solving a System of Equations using Elimination Method
This article will demonstrate how to solve the system of linear equations:
(1) (a-b)x + (a+b)y = a² - 2ab - b² (2) (a+b)x + (a+b)y = a² + b²
using the elimination method.
1. Identify the Variable to Eliminate
Observe that the coefficients of y are the same in both equations. Therefore, we can eliminate y by subtracting the equations.
2. Subtract the Equations
Subtracting equation (2) from equation (1) gives us:
(1) - (2): [(a-b)x + (a+b)y] - [(a+b)x + (a+b)y] = (a² - 2ab - b²) - (a² + b²)
Simplifying the equation:
-2bx = -2ab - 2b²
3. Solve for the Remaining Variable
Divide both sides of the equation by -2b (assuming b ≠ 0):
x = (2ab + 2b²) / 2b
Simplifying further:
x = a + b
4. Substitute the Value of x to Find y
Substitute the value of x (a + b) into either equation (1) or (2) to solve for y. Let's use equation (1):
(a-b)(a+b) + (a+b)y = a² - 2ab - b²
Expanding the left side:
a² - b² + (a+b)y = a² - 2ab - b²
Simplifying and solving for y:
(a+b)y = -2ab
y = -2ab / (a+b)
5. Solution
Therefore, the solution to the system of equations is:
x = a + b
y = -2ab / (a+b)
Note: This solution is valid only if b ≠ 0 and a ≠ -b.