+0 The system of equations
{xy}/{x + y} = 1,{xz}/{x + z} = 2, {yz}/{y + z} = 4
has exactly one solution. What is z in this solution?
I need help, I figured that x and y are both 2, but it's impossible.
+193 Here's how to find the solution for z:
Multiply each equation by its denominator:
xy = x + y (Equation 1)
xz = 2(x + z) (Equation 2)
yz = 4(y + z) (Equation 3)
Rearrange equations to isolate z:
From Equation 2: z = (xz - 2x) / 2 (Divide both sides by x)
Substitute this expression for z in Equation 3: yz = 4(y + (xz - 2x) / 2)
Simplify and solve for y:
yz = 2y + 2xz - 4x
Group terms with y: yz - 2y = 2xz - 4x
Factor out y: y(z - 2) = 2x(z - 2)
Divide both sides by (z - 2) and solve for y: y = 2x
Substitute y back into Equation 1:
xy = x + 2x
Combine like terms: xy = 3x
Divide both sides by x: y = 3
Combine information about y from steps 3 and 4:
We know y = 3 and y = 2x.
Therefore, 3 = 2x.
Solve for x: x = 3/2.
Substitute x and y back into the expression for z from step 2:
z = (x(3/2) - 2(3/2)) / 2
Simplify: z = (3/2 - 3) / 2
Therefore, z = -3/4.
Therefore, in the solution where the system of equations has exactly one solution, z is -3/4.
