If you add Natalie's age and Fred's age, the result is 44. If you add Fred's age to 4 times Natalie's age, the result is 83.

Write and solve a system of equations to find how old Fred and Natalie are.

1. Let [tex] N [/tex] be Natalie's age and [tex] F [/tex] be Fred's age.

2. Formulate the equations based on the given conditions:
- [tex] N + F = 44 [/tex]
- [tex] F + 4N = 83 [/tex]

3. Solve the system of equations to determine [tex] N [/tex] and [tex] F [/tex].

Fred is _ years old. Natalie is _ years old.

To solve the problem of finding Natalie's and Fred's ages, let's break down the information given and set up a system of equations.

1. Understand the Problem:
- If you add Natalie's age and Fred's age, the result is 44.
- This can be written as: [tex]$ N + F = 44 $[/tex]
- If you add Fred's age to 4 times Natalie's age, the result is 83.
- This can be written as: [tex]$ F + 4N = 83 $[/tex]

2. Set Up the Equations:
- We have two equations:
1. [tex]$ N + F = 44 $[/tex]
2. [tex]$ F + 4N = 83 $[/tex]

3. Solve the Equations:
a. From the first equation, express one variable in terms of the other. Let's solve for [tex]$ F $[/tex] in terms of [tex]$ N $[/tex]:
[tex]\[
F = 44 - N
\][/tex]

b. Substitute [tex]$ F = 44 - N $[/tex] into the second equation:
[tex]\[
(44 - N) + 4N = 83
\][/tex]

c. Simplify the equation:
[tex]\[
44 - N + 4N = 83
\][/tex]
[tex]\[
44 + 3N = 83
\][/tex]

d. Solve for [tex]$ N $[/tex]:
[tex]\[
3N = 83 - 44
\][/tex]
[tex]\[
3N = 39
\][/tex]
[tex]\[
N = 13
\][/tex]

4. Find Fred's Age:
- Use [tex]$ N = 13 $[/tex] in the expression for [tex]$ F $[/tex]:
[tex]\[
F = 44 - N = 44 - 13 = 31
\][/tex]

So, Natalie is 13 years old, and Fred is 31 years old.