Elimination Method to Solve Equations with Two Variables:

Elimination method is a technique in mathematics used to solve equations with two variables. It involves eliminating one of the variables in the two equations and solving for the other variable. Here is how to solve the given equations using elimination method:

Step 1: Multiply and Subtract

Multiply the first equation by b and the second equation by a. This will give:

abx + b^2y = b
abx + a^2y = 2ab

Now subtract the second equation from the first:

abx + b^2y - (abx + a^2y) = b - 2ab
b^2y - a^2y = b - 2ab

Factor out y on the left-hand side:

y(b^2 - a^2) = b - 2ab

Simplify the right-hand side:

y = (b - 2ab)/(b^2 - a^2)

Step 2: Solve for x

Now that we have solved for y, we can substitute this value into either equation to solve for x. Let's use the first equation:

ax + by = 1

Substitute y:

ax + ((b - 2ab)/(b^2 - a^2)) = 1

Multiply both sides by (b^2 - a^2):

a(b^2 - a^2)x + (b - 2ab) = b^2 - a^2

Simplify:

ab^2x - a^3x + b - 2ab = b^2 - a^2

Group like terms:

ab^2x - a^3x = b^2 - 3ab - a^2 + b

Factor out x:

x(ab^2 - a^3) = b^2 - 3ab - a^2 + b

Simplify:

x = (b^2 - 3ab - a^2 + b)/(ab^2 - a^3)

Therefore, the solution for the system of equations is:

x = (b^2 - 3ab - a^2 + b)/(ab^2 - a^3)
y = (b - 2ab)/(b^2 - a^2)