Determine the sum of the positive valued solution to the simultaneous equations:
xy=15, yz=35,zx=21
We can find the value of x by dividing equation 2 by equation 3:
$\frac{yz}{zx} = \frac{35}{21}$
$\frac{y}{z} = \frac{5}{3}$
Multiplying this equation by equation 1 gives us:
$\frac{y}{z} \cdot xy = \frac{5}{3} \cdot 15$
$y^2 = 25$
$y = 5$
Substituting this value of y into equation 2 gives us:
$z \cdot 5 = 35$
$z = 7$
And substituting this value of z into equation 1 gives us:
$x \cdot 5 \cdot 7 = 15$
$x = \frac{3}{7}$
The sum of the positive valued solutions is $5 + 7 + \frac{3}{7} = \boxed{\frac{119}{7}}$.
To find the sum of the positive valued solutions to the simultaneous equations xy = 15, yz = 35, and zx = 21, we need to solve for x, y, and z.
First, let's solve for x by dividing the equation zx = 21 by z:
x = 21/z
Now let's substitute this value of x into the equation xy = 15:
(21/z)y = 15
Next, we can solve for y by rearranging the equation:
y = (15z)/21 = (5z)/7
Now, let's substitute the value of y into the equation yz = 35:
(5z/7) * z = 35
Simplifying, we get:
5z^2/7 = 35
To solve for z, we multiply both sides of the equation by 7/5:
z^2 = 49
Taking the square root of both sides, we get:
z = ±7
Since we are looking for positive values, we can discard the negative solution. Thus, z = 7.
Now, let's substitute this value of z into the equation x = 21/z to find x:
x = 21/7 = 3
Finally, substituting the values of x and z into the equation y = (5z)/7, we get:
y = (5*7)/7 = 5
Therefore, the positive valued solutions to the simultaneous equations are x = 3, y = 5, and z = 7.
The sum of the positive valued solutions is:
3 + 5 + 7 = 15.
So, the sum of the positive valued solutions to the simultaneous equations is 15.
To determine the sum of the positive valued solutions to the simultaneous equations xy = 15, yz = 35, and zx = 21, we will solve this system of equations step by step.
Step 1: Solve for x in terms of y and z:
From the equation xy = 15, we can rearrange it to get x = 15/y.
Step 2: Substitute the expression for x into the equation zx = 21:
Replacing x with 15/y, we get (15/y)z = 21. To remove the fraction, we can multiply both sides of the equation by y, resulting in 15z = 21y. Rearranging this equation, we have z = 21y/15, which simplifies to z = 7y/5.
Step 3: Substitute the expressions for x and z into the equation yz = 35:
Plugging in the values of x and z into the equation, we get (15/y) * (7y/5) = 35. Simplifying further, we have 105y/5y = 35. Dividing both sides by 5, we find 21 = 35.
Step 4: Determine the value of y by solving the equation:
Since 21 = 35 is not a true statement, it means there is no positive valued solution for the variable y in this system of equations. Therefore, there is no positive valued solution to find the sum of.
In conclusion, there are no positive valued solutions to the simultaneous equations xy = 15, yz = 35, and zx = 21, and thus the sum of the positive valued solutions is undefined.