The sum of three numbers is 16. Two times the smallest is 2 less than the largest, while the sum of the largest and smallest is 11. Use a linear system in three variables to find the three numbers.
assume that x <= y <= z
Then we are told that
x+y+z = 16
2x = z-2
x+z = 11
Now just crank it out.
To solve this problem using a linear system, let's assign variables to represent the three numbers. Let's call the smallest number "x," the middle number "y," and the largest number "z."
Given the information in the problem, we can set up these equations:
Equation 1: x + y + z = 16 (sum of the three numbers is 16)
Equation 2: 2x = z - 2 (two times the smallest is 2 less than the largest)
Equation 3: x + z = 11 (sum of the largest and smallest is 11)
Now we have a system of three linear equations with three variables. To solve this system, we can use a method called substitution:
Step 1: Solve Equation 3 for z:
z = 11 - x
Step 2: Substitute z in Equation 2 with the value obtained from Step 1:
2x = (11 - x) - 2
2x = 9 - x
Step 3: Simplify Equation 2:
3x = 9
x = 3
Step 4: Substitute the value of x in Equation 1:
3 + y + z = 16
y + z = 13
Step 5: Substitute the value of x in Equation 3:
3 + z = 11
z = 8
Step 6: Substitute the values of x and z in Equation 4:
y + 8 = 13
y = 5
Therefore, the three numbers are:
x = 3
y = 5
z = 8
So, the smallest number is 3, the middle number is 5, and the largest number is 8.