The sum of three numbers is 9.the sum of twice the first number, 4times the second number, and 5times the third number is 25.the difference between 6times the first number and the second number is 34.find the three numbers.

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)