The sum of Joyce and Vangie's ages is 36. Twelve years ago, Vangie was twice as old as Joyce. Find their present ages. solve for x 

Let's define the problem using algebraic equations.

### Step 1: Define the Variables
- Let \( J \) be Joyce's current age.
- Let \( V \) be Vangie's current age.

### Step 2: Set Up the Equations
We have two pieces of information:
1. The sum of Joyce's and Vangie's ages is 36:
   \[
   J + V = 36
   \]
2. Twelve years ago, Vangie was twice as old as Joyce:
   \[
   V - 12 = 2 \times (J - 12)
   \]

### Step 3: Solve the System of Equations

#### From the first equation:
\[
V = 36 - J
\]

#### Substitute \( V = 36 - J \) into the second equation:
\[
(36 - J) - 12 = 2 \times (J - 12)
\]

#### Simplify the equation:
\[
36 - J - 12 = 2J - 24
\]
\[
24 - J = 2J - 24
\]

#### Add \( J \) and 24 to both sides:
\[
24 + 24 = 2J + J
\]
\[
48 = 3J
\]

#### Solve for \( J \):
\[
J = \frac{48}{3} = 16
\]

### Step 4: Find Vangie's Age
Substitute \( J = 16 \) back into the first equation:
\[
V = 36 - 16 = 20
\]

### Conclusion
- Joyce is 16 years old.
-

The sum of Joyce and Vangie's ages is 36. Twelve years ago, Vangie was twice as old as Joyce. Find their present ages. solve for x

Learn how to solve for the present ages of Joyce and Vangie given that their age sum is 36, and twelve years ago, Vangie was twice as old as Joyce. This problem demonstrates how to set up and solve a system of linear equations, suitable for students in Grades 6-8.

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