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dimulka [17.4K]
3 years ago
12

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.T

he difference between 6times the first number and the second number is 34.Find the three numbers.
Mathematics
1 answer:
Karo-lina-s [1.5K]3 years ago
5 0

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)

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