Age problem worksheet for solving systems of equations.

To solve the age problems using a system of equations, we will follow these steps for each problem:

---

Problem 1:

Statements:
- Eduardo is 6 years older than Tanaka.
- The sum of their ages is 28.

#### Step 1: Define variables
Let:
- $ E $ = Eduardo's age
- $ T $ = Tanaka's age

#### Step 2: Translate statements into equations
1. "Eduardo is 6 years older than Tanaka":
\[
E = T + 6
\]

2. "The sum of their ages is 28":
\[
E + T = 28
\]

#### Step 3: Solve the system of equations
We have:
\[
\begin{cases}
E = T + 6 \\
E + T = 28
\end{cases}
\]

Substitute $ E = T + 6 $ into the second equation:
\[
(T + 6) + T = 28
\]

Simplify:
\[
2T + 6 = 28
\]

Subtract 6 from both sides:
\[
2T = 22
\]

Divide by 2:
\[
T = 11
\]

Now, substitute $ T = 11 $ back into $ E = T + 6 $:
\[
E = 11 + 6 = 17
\]

#### Step 4: Verify the solution
- Eduardo's age ($ E $) = 17
- Tanaka's age ($ T $) = 11
- Check the sum: $ E + T = 17 + 11 = 28 $ (correct)
- Check the age difference: $ E - T = 17 - 11 = 6 $ (correct)

Thus, the solution is:
\[
\boxed{E = 17, T = 11}
\]

---

Problem 2:

Statements:
- Eduardo is 1 year older than Clarieta.
- The sum of their ages is 23.

#### Step 1: Define variables
Let:
- $ E $ = Eduardo's age
- $ C $ = Clarieta's age

#### Step 2: Translate statements into equations
1. "Eduardo is 1 year older than Clarieta":
\[
E = C + 1
\]

2. "The sum of their ages is 23":
\[
E + C = 23
\]

#### Step 3: Solve the system of equations
We have:
\[
\begin{cases}
E = C + 1 \\
E + C = 23
\end{cases}
\]

Substitute $ E = C + 1 $ into the second equation:
\[
(C + 1) + C = 23
\]

Simplify:
\[
2C + 1 = 23
\]

Subtract 1 from both sides:
\[
2C = 22
\]

Divide by 2:
\[
C = 11
\]

Now, substitute $ C = 11 $ back into $ E = C + 1 $:
\[
E = 11 + 1 = 12
\]

#### Step 4: Verify the solution
- Eduardo's age ($ E $) = 12
- Clarieta's age ($ C $) = 11
- Check the sum: $ E + C = 12 + 11 = 23 $ (correct)
- Check the age difference: $ E - C = 12 - 11 = 1 $ (correct)

Thus, the solution is:
\[
\boxed{E = 12, C = 11}
\]

---

Problem 3:

Statements:
- Asphondy is 2 years older than Tanaka.
- The sum of their ages is 4.

#### Step 1: Define variables
Let:
- $ A $ = Asphondy's age
- $ T $ = Tanaka's age

#### Step 2: Translate statements into equations
1. "Asphondy is 2 years older than Tanaka":
\[
A = T + 2
\]

2. "The sum of their ages is 4":
\[
A + T = 4
\]

#### Step 3: Solve the system of equations
We have:
\[
\begin{cases}
A = T + 2 \\
A + T = 4
\end{cases}
\]

Substitute $ A = T + 2 $ into the second equation:
\[
(T + 2) + T = 4
\]

Simplify:
\[
2T + 2 = 4
\]

Subtract 2 from both sides:
\[
2T = 2
\]

Divide by 2:
\[
T = 1
\]

Now, substitute $ T = 1 $ back into $ A = T + 2 $:
\[
A = 1 + 2 = 3
\]

#### Step 4: Verify the solution
- Asphondy's age ($ A $) = 3
- Tanaka's age ($ T $) = 1
- Check the sum: $ A + T = 3 + 1 = 4 $ (correct)
- Check the age difference: $ A - T = 3 - 1 = 2 $ (correct)

Thus, the solution is:
\[
\boxed{A = 3, T = 1}
\]

---

Problem 4:

Statements:
- Johnathan is 5 years older than Ji.
- The sum of their ages is 9.

#### Step 1: Define variables
Let:
- $ J_1 $ = Johnathan's age
- $ J_2 $ = Ji's age

#### Step 2: Translate statements into equations
1. "Johnathan is 5 years older than Ji":
\[
J_1 = J_2 + 5
\]

2. "The sum of their ages is 9":
\[
J_1 + J_2 = 9
\]

#### Step 3: Solve the system of equations
We have:
\[
\begin{cases}
J_1 = J_2 + 5 \\
J_1 + J_2 = 9
\end{cases}
\]

Substitute $ J_1 = J_2 + 5 $ into the second equation:
\[
(J_2 + 5) + J_2 = 9
\]

Simplify:
\[
2J_2 + 5 = 9
\]

Subtract 5 from both sides:
\[
2J_2 = 4
\]

Divide by 2:
\[
J_2 = 2
\]

Now, substitute $ J_2 = 2 $ back into $ J_1 = J_2 + 5 $:
\[
J_1 = 2 + 5 = 7
\]

#### Step 4: Verify the solution
- Johnathan's age ($ J_1 $) = 7
- Ji's age ($ J_2 $) = 2
- Check the sum: $ J_1 + J_2 = 7 + 2 = 9 $ (correct)
- Check the age difference: $ J_1 - J_2 = 7 - 2 = 5 $ (correct)

Thus, the solution is:
\[
\boxed{J_1 = 7, J_2 = 2}
\]

---

Final Answers:

1. $\boxed{E = 17, T = 11}$
2. $\boxed{E = 12, C = 11}$
3. $\boxed{A = 3, T = 1}$
4. $\boxed{J_1 = 7, J_2 = 2}$