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Find the perimeter of the equilateral triangle

The triangle in the image is an equilateral triangle, which means all sides are equal in length. We are given the following algebraic expressions for each side:

- Left side: \( 2y \)
- Right side: \( 2y \)
- Bottom side: \( 3x - z + 20 \)
- Top side: \( x + y - 10 \)
- Right side (another expression): \( 2z - y \)

Since all sides must be equal in an equilateral triangle, we can set up the following system of equations based on the equality of sides:

1. \( 2y = x + y - 10 \)
2. \( 2y = 3x - z + 20 \)
3. \( 2y = 2z - y \)

We will now solve this system of equations to find \( x \), \( y \), and \( z \), and use the value of any side to calculate the perimeter of the triangle.

### Step 1: Solve equation (1)
\[
2y = x + y - 10
\]
Subtract \( y \) from both sides:
\[
y = x - 10
\]
Thus, \( x = y + 10 \).

### Step 2: Solve equation (3)
\[
2y = 2z - y
\]
Add \( y \) to both sides:
\[
3y = 2z
\]
So, \( z = \frac{3y}{2} \).

### Step 3: Substitute \( x = y + 10 \) and \( z = \frac{3y}{2} \) into equation (2)
\[
2y = 3x - z + 20
\]
Substitute \( x = y + 10 \) and \( z = \frac{3y}{2} \):
\[
2y = 3(y + 10) - \frac{3y}{2} + 20
\]
Simplify:
\[
2y = 3y + 30 - \frac{3y}{2} + 20
\]
Combine terms:
\[
2y = 3y - \frac{3y}{2} + 50
\]
To simplify further, let's combine \( 3y - \frac{3y}{2} \). This becomes:
\[
3y - \frac{3y}{2} = \frac{6y}{2} - \frac{3y}{2} = \frac{3y}{2}
\]
So the equation becomes:
\[
2y = \frac{3y}{2} + 50
\]
Multiply everything by 2 to eliminate the fraction:
\[
4y = 3y + 100
\]
Subtract \( 3y \) from both sides:
\[
y = 100
\]

### Step 4: Calculate \( x \) and \( z \)
From \( x = y + 10 \):
\[
x = 100 + 10 = 110
\]
From \( z = \frac{3y}{2} \):
\[
z = \frac{3(100)}{2} = 150
\]

### Step 5: Find the perimeter
Now, substitute \( y = 100 \) into one of the side expressions, say \( 2y \):
\[
2y = 2(100) = 200
\]
Since all sides are equal, the perimeter of the equilateral triangle is:
\[
\text{Perimeter} = 3 \times 200 = 600
\]

The perimeter of the triangle is **600 units**.

Would you like a deeper explanation or further details? Here are five related questions for practice:

1. How do you determine the perimeter of an equilateral triangle if all sides are given as algebraic expressions?
2. Can you solve a similar triangle problem if the expressions for each side involve different variables?
3. What happens if you change one of the equations in the system—how does that affect the perimeter?
4. How would you handle this problem if it were not an equilateral triangle but an isosceles triangle instead?
5. What are some alternative methods to solve systems of equations involving geometric figures?

**Tip**: Always check for consistency in geometry problems involving algebraic expressions by ensuring the properties of the shape (e.g., all sides equal in an equilateral triangle) are met.

Find the perimeter of the equilateral triangle.

Learn how to find the perimeter of an equilateral triangle with algebraic expressions for its sides. This problem involves solving a system of equations and applying geometric properties. Suitable for students in Grades 9-12.

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