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 = Smallest #.

4X = Largest @.
X+ 8X = 18.
9X = 18,

X = 2.
4X = 4*2 = 8.
14 - 10 = 4 = Middle #.

A. 1 unknown: X.

B. X = Smallest #.
4X = Largest #.
X + 8X = 18.

C.

A. In this problem, we are dealing with 3 unknowns, as indicated by the fact that we need to find 3 numbers. The unknowns in this case are the three numbers.

B. To obtain the equations, we can use the information given in the problem statement.

1. The sum of the three numbers is 14.
This can be expressed as:
x + y + z = 14 ----- (Equation 1)

2. The largest number is 4 times the smallest.
This can be expressed as:
z = 4x ----- (Equation 2)

3. The sum of the smallest number and twice the largest number is 18.
This can be expressed as:
x + 2z = 18 ----- (Equation 3)

C. Now, let's solve the system of linear equations using matrices:

Step 1: Rearrange Equations 2 and 3 to eliminate 'z' variable.
Equation 2: z = 4x
Equation 3: x + 2z = 18
Multiply Equation 3 by (-2) to get -2x - 4z = -36.

Step 2: Rewrite the equations in matrix form.
The augmented matrix representing this system of equations is:

[ 1 1 1 | 14 ] ----- (Matrix A)
[ 4 0 -1 | 0 ] ----- (Matrix B)
[ 1 0 2 | 18 ] ----- (Matrix C)

Step 3: Perform row operations to find the solution.
- Multiply Row 1 by (-4) and add it to Row 2: Row 2 - 4 * Row 1 = [0 -4 -5 | -4]
- Multiply Row 1 by (-1) and add it to Row 3: Row 3 - Row 1 = [-1 -1 1 | 4]
- Multiply Row 2 by (-1/4): (1/4) * Row 2 = [0 1 5/4 | 1]

The new augmented matrix after these row operations is:

[ 1 1 1 | 14 ]
[ 0 -4 -5 | -4 ]
[ 0 -1 1 | 4 ]

Step 4: Continue row operations to simplify the matrix.
- Multiply Row 2 by (-1/4): (1/4) * Row 2 = [0 1 5/4 | 1]
- Multiply Row 2 by -1 and add it to Row 3: Row 3 - Row 2 = [0 0 1 | 3]

The new augmented matrix after these row operations is:

[ 1 1 1 | 14 ]
[ 0 1 5/4 | 1 ]
[ 0 0 1 | 3 ]

Step 5: Back-substitution to find the values of x, y, and z.
Equation 3: z = 3
Equation 2: y + (5/4)z = 1
Substitute z with 3 in Equation 2: y + (5/4) * 3 = 1
y + (15/4) = 1
y = 1 - (15/4)
y = -11/4

Substitute z = 3 and y = -11/4 in Equation 1: x + y + z = 14
x + (-11/4) + 3 = 14
x - 11/4 + 12/4 = 14
x + 1/4 = 14
x = 14 - 1/4
x = 55/4

So, the three numbers are:
x = 55/4
y = -11/4
z = 3

Therefore, the three numbers are 55/4, -11/4, and 3.