Solve for x and y using elimination
ax+by=c
bx+ay=c
This question has been confusing me for a long time, and I can't seem to find the answer by solving using elimination.
Thanks for helping me :)
multiply #1 by b and #2 by a, and you have
abx+b^2y = bc
abx+a^2y = ac
now subtract
(b^2-a^2)y = (b-a)c
y = c/(a+b)
and use that to find x. Or, start all over and multiply by a and b to remove the y terms.
To solve for x and y using elimination, we'll use a method called addition or subtraction method.
Given the equations:
ax + by = c
bx + ay = c
Step 1: Multiply the first equation by b, and the second equation by a:
abx + b^2y = bc
abx + a^2y = ac
Step 2: Subtract the second equation from the first equation to eliminate the x terms:
abx - abx + b^2y - a^2y = bc - ac
(b^2 - a^2)y = bc - ac
(b + a)(b - a)y = c(b - a)
Step 3: Divide both sides of the equation by (b + a)(b - a):
y = (c(b - a))/(b + a)(b - a)
y = c/(b + a)
Step 4: Substitute the value of y into one of the original equations, let's choose the first one:
ax + b(c/(b + a)) = c
ax(b + a) + bc = c(b + a)
abx + a^2x + bc = bc + ac
abx + a^2x = ac
Step 5: Factor out x from the equation:
x(ab + a^2) = ac
Step 6: Divide both sides of the equation by (ab + a^2):
x = ac/(ab + a^2)
So, the solution for x is x = ac/(ab + a^2) and the solution for y is y = c/(b + a).