system of linear equation

The sum of 3 numbers is 14
the largest is 4 times the smallest
the sum of the smallest and twice the largest is 18
what are the 3 numbers? show work?

A. how many unknowns you have?
-how many equations need to be solve for that many unknown?

B. what are the equations?

C. used the system of equations using matrices.

x+y+z=14

z=4x
x+2z=18
=================
x+y+4x = 14 or 5x+y = 14
x+8x = 18
or x = 2
so
y = 14 - 10 = 4
z = 4*2=8
3 unknowns, three equations
I did it with substitution, however can form augmented matrix and use Gauss Jordan:
+1 1 1 14
-4 0 1 0
+1 0 2 18

then plug and chug

post it.

matix solution

|1 1 1 | [x y z]'= [14 0 18]'
|4 0 1 |
|1 0 2 |

H*[x y z]' = [14 0 18]' (' stands for transpose; in this case makes row vector a column vector)

Answer

inv(H)*H*[x y z]'= inv(H)*[14 0 18]'

[x y z] = inv(H)*[14 0 18]'

inverting a 3x3 matrix

inv(H)=1/|H|*adj(H) |H|=det(H)

adj(H)=

|+(0-0) -(-8-1) +(0-0)|'
|-(2-0) -(2-1) -(0-1)|
|+(1-0) -(1+4) +(0+4)|

|0 9 0|'
|-2 -1 1|
|1 -5 4|

=|0 -2 1|
|9 -1 -5|
|0 1 4|

det(H)= use the second column
since it has 2 zeros it makes
it easy.

det(H)=-1(-8-1)+0*(2-1)-0*(1+4)=9

inv(H)=1/9*|0 -2 1|
|9 -1 -5|
|0 1 4|

[x y z]'= inv(H)*[14 0 18]'

=[18/9 (9*14-5*18)/9 (18*4)/9]'
=[ 2 4 8 ]'

hope this helps

Let the no be x,y,z

x+y+z=14
z=4x
x+2z=18
from eqn 3
x+2(4x)=18
x+8x=18
9x=18
x=2,sub into eqn 2
z=4(2)
z=8
y=4.

A. You have three unknowns, namely the three numbers mentioned in the problem.

B. To solve the problem, you need three equations based on the given information. Let's assign variables to the three numbers:

Let the smallest number be x.
The largest number is 4 times the smallest, so the largest number is 4x.
The sum of the smallest number and twice the largest number is 18, which can be expressed as x + 2(4x) = 18.

The sum of the three numbers is 14, which can be written as x + 4x + 14 = 14.

Now, we have two equations:
1. x + 4x + 14 = 14
2. x + 2(4x) = 18

C. We can rewrite these equations in the form of a system of linear equations using matrices.

Equation 1: x + 4x + 14 = 14
This equation can be rearranged as follows:
5x = 0
Or in matrix form:
[5x = 0]

Equation 2: x + 2(4x) = 18
Rearranging this equation gives us:
9x = 18
In matrix form:
[9x = 18]

Now, we can solve this system of equations using matrices.

Augmented matrix form of the system:
[5 | 0]
[9 | 18]

To solve it, we can perform row operations to obtain the solution:
1. Divide Row 1 by 5:
[1 | 0]
[9 | 18]

2. Subtract 9 times Row 1 from Row 2:
[1 | 0]
[0 | 18]

So, the solution to this system of equations is x = 0 and 18.

Now substituting these values back into the original equations we can find the values of the smallest, largest, and third numbers.

From equation 1, substitute x = 0:
0 + 4(0) + 14 = 14
14 = 14

From equation 2, substitute x = 0:
0 + 2(4 * 0) = 18
0 = 18

So, the values obtained are not valid solutions. Therefore, there is no unique solution to this system of linear equations.