Determine the sum of the positive valued solution to the simultaneous equations:
xy = 15, yz = 35,zx = 21
To determine the sum of the positive valued solution to the simultaneous equations, let's solve them step by step.
Step 1: Solve for x in terms of y and z.
From the equation xy = 15, we get: x = 15/y.
Step 2: Substitute the value of x into the equation zx = 21.
Substituting x = 15/y into zx = 21, we have: (15/y)z = 21.
Solving for z, we get: z = 21y/15.
Step 3: Substitute the values of x and z into the equation yz = 35.
Substituting x = 15/y and z = 21y/15 into yz = 35, we have: (15/y) * (21y/15) = 35.
Simplifying the equation, we get: 21 = 35.
This equation has no solution, which means that there are no positive valued solutions to the given simultaneous equations.
Therefore, the sum of the positive valued solution(s) is undefined.
We can solve this system of equations by multiplying all three equations together:
(xy)(yz)(zx) = 15 * 35 * 21
xyz * xyz * xyz = 15 * 35 * 21
(xyz)^2 = 15 * 35 * 21
xyz = √(15 * 35 * 21)
We know that xyz is positive because all three variables (x, y, and z) are positive.
Now, if we divide each equation by the corresponding variable, we get:
xy / x = y
yz / y = z
zx / z = x
Substituting the given values, we have:
y = 15 / x
z = 35 / y
x = 21 / z
Substituting these values into the equation xyz = √(15 * 35 * 21), we get:
(21 / z) * (15 / (15 / x)) * (35 / (35 / y)) = √(15 * 35 * 21)
21 * x * 35 * y * 15 * z = √(15 * 35 * 21)^2
(21 * 35 * 15) * xyz = 15 * 35 * 21
xyz = 1
Therefore, the positive valued solution to this system of equations is xyz = 1.
The sum of the positive valued solution is 1 + 1 + 1 = 3.
To find the sum of the positive valued solutions to the simultaneous equations, we need to first solve the equations to find the individual values of x, y, and z.
Let's start with the first equation: xy = 15.
From this equation, we can express x in terms of y by dividing both sides by y: x = 15/y.
Now let's move on to the second equation: yz = 35.
We can express z in terms of y by dividing both sides by y: z = 35/y.
Finally, let's look at the third equation: zx = 21.
We can substitute the values of x and z that we found earlier into this equation:
(15/y)(35/y) = 21.
Simplifying this equation, we get:
525/y^2 = 21.
To further simplify, we can multiply both sides by y^2:
525 = 21y^2.
Now we can solve for y by dividing both sides by 21 and taking the square root:
y^2 = 25.
Taking the positive square root, we get:
y = 5.
Now that we have the value of y, we can substitute it back into the equations to find the values of x and z.
From x = 15/y, we get x = 3.
From z = 35/y, we get z = 7.
We have found that x = 3, y = 5, and z = 7.
To determine the sum of the positive valued solutions, we simply add them up:
3 + 5 + 7 = 15.
Therefore, the sum of the positive valued solutions to the simultaneous equations is 15.