if x, y, and z are positive numbers, with xy = 24, xz = 48, and yz = 72, what is x + y + x? please explain your answer too. thankyou.
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To solve for the values of x, y, and z, we can use a system of equations. Let's solve it step by step:
1. We are given that xy = 24, xz = 48, and yz = 72.
2. To eliminate z, divide the second equation by the first equation: (xz)/(xy) = 48/24
This simplifies to z = 2. So, we have found the value of z.
3. Substitute z = 2 into the first equation: xy = 24
Rearranging this equation will give us y in terms of x: y = 24/x.
4. Substitute the value of z in the third equation: yz = 72
Substituting y = 24/x will give us (24/x) * 2 = 72.
Simplifying the equation further, we get 48/x = 72.
5. Solve for x by isolating it. Multiply both sides of the equation by x: 48 = 72x.
Divide both sides by 72: x = 48/72.
Simplifying 48/72 gives us x = 2/3.
6. Substitute the value of x back into the first equation: xy = 24
Putting x = 2/3, we have (2/3)y = 24.
Solve for y by multiplying both sides by 3/2: y = (24 * 3)/2.
Simplifying further, we get y = 36.
7. We can find y using the second equation: xz = 48.
Substituting x = 2/3, we have (2/3)z = 48.
Solve for z by multiplying both sides by 3/2: z = (48 * 3)/2.
Simplifying further, we get z = 72.
Therefore, we have found the values of x = 2/3, y = 36, and z = 72.
To find x + y + x:
Substituting the values we found, x = 2/3, y = 36, and z = 72, into the expression x + y + x, we get (2/3) + 36 + (2/3).
Adding the fractions, we get (4/3) + 36.
Now, convert 4/3 into an improper fraction by dividing 4 by 3, which gives us 1 and a remainder of 1.
So, (4/3) is equivalent to 1 and 1/3.
Adding 1 and 1/3 to 36 gives us a final result of 37 and 1/3.
Therefore, x + y + x = 37 and 1/3.