solve the following:
ax + by = a-b
bx - ay = a+b
find the value of x and y
This is not perfect solution
To solve the given system of equations, we will use the method of elimination. Here are the steps to follow:
Step 1: Multiply both equations by the necessary coefficients to make the coefficients of x or y in one of the equations cancel out when added or subtracted.
Let's multiply equation 1 by b and equation 2 by a to eliminate the x term:
Equation 1 (after multiplication): abx + b^2y = (a - b)b
Equation 2 (after multiplication): abx - a^2y = (a + b)a
Step 2: Subtract equation 2 from equation 1 to eliminate the x term.
(abx + b^2y) - (abx - a^2y) = (a - b)b - (a + b)a
abx - abx + b^2y + a^2y = ab - b^2 - a^2
Simplifying the equation, we get:
b^2y + a^2y = ab - b^2 - a^2
Step 3: Combine like terms.
(b^2 + a^2)y = ab - b^2 - a^2
Step 4: Divide both sides of the equation by (b^2 + a^2).
y = (ab - b^2 - a^2) / (b^2 + a^2)
Now that we have the value of y, we can substitute it back into either equation 1 or equation 2 to find the value of x.
Let's substitute it into equation 1:
ax + by = a - b
ax + b((ab - b^2 - a^2) / (b^2 + a^2)) = a - b
Now, solve for x by isolating it:
ax + (ab - b^2 - a^2)b / (b^2 + a^2) = a - b
ax(b^2 + a^2) + (ab - b^2 - a^2)b = (a - b)(b^2 + a^2)
ab^2x + a^3x + ab^2 - b^3 - ab^2 - a^3 = a(b^2 + a^2) - b(b^2 + a^2)
Simplifying the equation:
ab^2x + a^3x - b^3 = ab^2 + a^3 - b^3 - ab^2 - ab^2
ab^2x + a^3x - b^3 = a^3 - b^3
Step 5: Divide both sides of the equation by (a^3 - b^3):
x = (a^3 - b^3) / (ab^2 + a^3 - b^3)
Now, replace the values of a and b in the expression for x and y to get the final solution.
abx + b^2y = b(a-b)
abx - a^2y = a(a+b)
(b^2+a^2)y = -(a^2+b^2)
y = -1
similarly,
x = 1