Answer:
x = 6, y = 2, z = 1
Step-by-step explanation:
This is a system of 3 equations with 3 unknowns. We need to first write the equations given the information and then solve them.
The first equation is the sum of 3 numbers, all unknown, is 9:
x + y + z = 9
The second equation is again a sum:
2x + 4y + 5z = 25
The third equation is the difference of only the first 2 numbers:
6x - y = 34
We will start with that last equation and solve it for y:
-y = -6x + 34 so
y = 6x - 34
Now we will go back to the first 2 equations and sub that 6x-34 in for each y. The first equation then becomes:
x + 6x - 34 + z = 9 and 7x + z = 43
The second equation then becomes:
2x + 4(6x - 34) + 5z = 25 and 26x + 5z = 161
We will solve those 2 bold equations by addition/elimination:
7x + z = 43
26x + 5z = 161
Multiply the first equation through by -5 to get rid of the z's. That equation then becomes:
-35x - 5z = -215
26x + 5z = 161
Adding straight down the columns gives you
-9x = -54 so
x = 6
Now we can plug that x value of 6 into any equation that has an x in it:
26(6) + 5z = 161 and
156 + 5z = 161 and
5z = 5 so
z = 1.
We can use the x value only again in the equation we solved for y in the beginning:
y = 6(6) - 34 so
y = 36 - 34 and
y = 2
The solution to this system in coordinate form is
(6, 2, 1)
