Solving a System of Linear Equations with Two Variables
This article will discuss how to solve a system of linear equations with two variables, represented by the following equations:
(a + 2b)x + (2a  b)y = 2
(a  2b)x + (2a + b)y = 3
We can solve this system using various methods, including substitution, elimination, and matrix methods. Here, we will focus on the elimination method.
Elimination Method
The elimination method aims to eliminate one of the variables by manipulating the equations to create opposite coefficients for that variable. Then, by adding the equations together, we can solve for the remaining variable.
Step 1: Identify a variable to eliminate
In this system, we can easily eliminate either 'x' or 'y' by multiplying the equations by appropriate constants. Let's choose to eliminate 'x'.
Step 2: Multiply equations to get opposite coefficients for 'x'

Multiply the first equation by (a  2b): (a  2b)(a + 2b)x + (a  2b)(2a  b)y = 2(a  2b)

Multiply the second equation by (a + 2b): (a + 2b)(a  2b)x  (a + 2b)(2a + b)y = 3(a + 2b)
Step 3: Add the modified equations together
Notice that the 'x' terms now have opposite coefficients and will cancel out when added. We are left with:
(a  2b)(2a  b)y  (a + 2b)(2a + b)y = 2(a  2b)  3(a + 2b)
Step 4: Simplify and solve for 'y'
Expanding and simplifying the equation:
(4a^2  5ab + 2b^2)y  (4a^2 + 3ab + 2b^2)y = a  8b 8aby = a  8b y = (a + 8b) / 8ab
Step 5: Substitute the value of 'y' into one of the original equations and solve for 'x'
Let's substitute 'y' into the first original equation:
(a + 2b)x + (2a  b)(a + 8b) / 8ab = 2
Simplifying and solving for 'x':
(a + 2b)x + (2a^2 + 15ab  8b^2) / 8ab = 2 (a + 2b)x = 2  (2a^2 + 15ab  8b^2) / 8ab (a + 2b)x = (16ab  2a^2  15ab + 8b^2) / 8ab (a + 2b)x = (ab + 8b^2  2a^2) / 8ab x = (ab + 8b^2  2a^2) / 8ab(a + 2b)
Solution
Therefore, the solution to the system of equations is:
 x = (ab + 8b^2  2a^2) / 8ab(a + 2b)
 y = (a + 8b) / 8ab
This solution can be expressed in terms of 'a' and 'b', and it represents the intersection point of the two linear equations when graphed.