The sum of three numbers is 102. The first number is 6 more than the second. The third number is 4 times the second. What are the numbers?

1st number: 22

2nd number: 16

3rd number: 64

Step-by-step explanation:

x + y + z = 102

x = 6 + y

z = 4y

plug that in!

(6 + y) + y + (4y) = 102

get rid of the parenthesis (I added them so you could see what I was replacing) and add like terms.

6 + 6y = 102

move 6 to the other side.

6y = 96

y = 16

now that you know one number, you can solve the equations for the rest!

x = 6 + (16)

z = 4(16)

x = 22

y = 16

z = 64

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.