x%2B(a%2Bb)y%3D2a%5E2-2b%5E2-(a%2Bb)(x%2By)%3D4ab-by-elimination-method.jpeg "(a-b)x+(a+b)y=2a^2-2b^2 (a+b)(x+y)=4ab By Elimination Method")
Solving a System of Linear Equations by Elimination Method
This article will demonstrate how to solve the following system of linear equations using the elimination method:
(1) (a-b)x + (a+b)y = 2a^2 - 2b^2 (2) (a+b)(x+y) = 4ab
1. Simplify the Equations
- Equation (2): Expand the left side of the equation: (a+b)x + (a+b)y = 4ab
2. Eliminate One Variable
Notice that the coefficients of 'x' and 'y' in the simplified equation (2) are the same as in equation (1). To eliminate 'x', we can multiply equation (2) by -(a-b) and add it to equation (1):
- Equation (1) * 1: (a-b)x + (a+b)y = 2a^2 - 2b^2
- Equation (2) * -(a-b): -(a^2-b^2)x - (a^2-b^2)y = -4ab(a-b)
Adding these two equations:
- (2b^2)y = 2a^2 - 2b^2 - 4ab(a-b)
3. Solve for the Remaining Variable
Simplify and solve for 'y':
- y = (a^2 - b^2 - 2ab(a-b)) / b^2
- y = (a^2 - b^2 - 2a^2b + 2ab^2) / b^2
- y = (a^2 - 2a^2b + 2ab^2 - b^2) / b^2
- y = (a^2(1 - 2b) + b^2(2a - 1)) / b^2
4. Substitute and Solve for the Other Variable
Substitute the value of 'y' found above into either equation (1) or (2) to solve for 'x'. For simplicity, let's use the simplified equation (2):
- (a+b)x + (a+b) * [(a^2(1 - 2b) + b^2(2a - 1)) / b^2] = 4ab
Simplify and solve for 'x':
- (a+b)x + (a^2(1 - 2b) + b^2(2a - 1)) / b = 4ab
- (a+b)bx + (a^2(1 - 2b) + b^2(2a - 1)) = 4ab^2
- (a+b)bx = 4ab^2 - (a^2(1 - 2b) + b^2(2a - 1))
- (a+b)bx = 4ab^2 - a^2 + 2a^2b - 2ab^2 + b^2
- (a+b)bx = 2a^2b - a^2 + b^2
- x = (2a^2b - a^2 + b^2) / (b(a+b))
5. The Solution
Therefore, the solution to the system of equations is:
- x = (2a^2b - a^2 + b^2) / (b(a+b))
- y = (a^2(1 - 2b) + b^2(2a - 1)) / b^2
This solution holds true for all values of 'a' and 'b' except for cases where the denominator in the expressions for 'x' and 'y' becomes zero (b=0 or a+b=0).