1/x = 1/y + 1/z
solve for Z
1/z=1/x + 1/y= (y+x)/xy
z= xy/(x+y)
oops, sign problem
1/z=1/x -1/y= (y-x)/xy
z= xy/(y-x)
check my work
To solve for z in the equation 1/x = 1/y + 1/z, we can follow these steps:
Step 1: Start by finding a common denominator between y and z. The common denominator can be found by simply multiplying y and z.
Step 2: Rewrite the right side of the equation with the common denominator. This gives us 1/y + 1/z = (z + y) / (y * z).
Step 3: Substitute the right side of the equation with the rewritten expression. Therefore, we have:
1/x = (z + y) / (y * z)
Step 4: Multiply both sides of the equation by x to eliminate the fraction on the left side:
x * 1/x = x * (z + y) / (y * z)
This simplifies to:
1 = (x * (z + y)) / (y * z)
Step 5: Multiply both sides of the equation by y * z to get rid of the fraction:
1 * (y * z) = (x * (z + y)) * (y * z) / (y * z)
Simplifying further, we have:
y * z = x * (z + y)
Step 6: Distribute x on the right side of the equation:
y * z = x * z + x * y
Step 7: Rearrange the equation by moving x * z to the left side:
y * z - x * z = x * y
Simplifying further, we have:
(z * (y - x)) = x * y
Step 8: Finally, solve for z by dividing both sides of the equation by (y - x):
z = (x * y) / (y - x)
Therefore, the solution for z is z = (x * y) / (y - x).