The sum of three numbers is 79. the second number is 5 times the first, while the third is 16 more than the first. find the numbers.

Let the 1st number be x; 2nd number be y; 3rd number be z.

x + y + z = 79

x = number we are looking for.
y = x * 5 ==> 5 times the first
z = x + 16 ==> 16 more than the first

Therefor,

x + (x * 5) + (x+16) = 79

1st step, multiply the 2nd number: x * 5 = 5x

x + 5x + x + 16 = 79

Add all like numbers:

7x + 16 = 79

To get x, transfer 16 to the other side and change its sign from positive to negative.

7x = 79 - 16
7x = 63

To get x, divide both sides by 7

7x/7 = 63/7
x = 9

To check. Substitute x by 9.

x + (x * 5) + (x+16) = 79

9 + (9 * 5) + (9 + 16) = 79
9 + 45 + 25 = 79
79 = 79 equal. value of x is correct.

This problem can be solved with a system of equations.
The variables we will use to solve this will be x, y, and z.
x will represent the first number.
y will represent the second number.
z will represent the third number.

The first equation in the system of equations we need to solve this is that the sum of all 3 numbers, x, y, and z, is 79.
x + y + z = 79

We are told the second number is 5 times greater than the first number.
Since y is the second number and x is the first number, this means y = 5x.

The last equation we'll be using to solve for the numbers is z = x + 16. This is because the third number, z, is 16 more, meaning plus, the first number, x.

{ x + y + z = 79
{ y = 5x
{ z = x + 16
Since y and z are already isolated, we can plug in the expressions equal to them in the first equation, x + y + z = 79.
x + (5x) + (x + 16) = 79

If we combine like terms then isolate the variable, we can solve for x. Then we can plug the value of x into the other two equations to solve for y and z.

Combining like terms:
x + (5x) + (x + 16) = 79
The like terms are x, 5x, and x.
x + 5x + x = 7x
7x + 16 = 79

Isolating the variable:
7x + 16 = 79
First, subtract 16 from both sides. Then, divide both sides by 7.
7x + 16 - 16 = 7x
79 - 16 = 63
7x / 7 = x
63 / 7 = 9
x = 9

Now that we know the value of x, we can plug it into the other two equations and solve for y and z.
Recall that y = 5x and z = x + 16.

Solving for y:
y = 5x
x = 9
y = 5(9)
5 • 9 = 45
y = 45

Solving for z:
z = x + 16
x = 9
z = 9 + 16
9 + 16 = 25
z = 25

Answers:
x = 9
y = 45
z = 25

Hope this helps!