The sides of an equilateral triangles are given by x+ 3y , 3x+ 2y - 2 and ` 4x + (1)/(2) y+ 1`

Answer

To solve the problem, we need to find the values of \( x \) and \( y \) such that the three expressions for the sides of the equilateral triangle are equal. The sides of the triangle are given as:

1. \( a = x + 3y \)
2. \( b = 3x + 2y - 2 \)
3. \( c = 4x + \frac{1}{2}y + 1 \)

Since it is an equilateral triangle, we have:

\[ a = b = c \]

### Step 1: Set up the equations

We will first set \( a = b \):

\[ x + 3y = 3x + 2y - 2 \]

### Step 2: Rearrange the equation

Rearranging the equation gives:

\[ x + 3y - 3x - 2y + 2 = 0 \]

This simplifies to:

\[ -2x + y + 2 = 0 \]

Rearranging further, we get:

\[ 2x - y = 2 \quad \text{(Equation 1)} \]

### Step 3: Set up the second equation

Next, we set \( b = c \):

\[ 3x + 2y - 2 = 4x + \frac{1}{2}y + 1 \]

### Step 4: Rearrange the second equation

Rearranging gives:

\[ 3x + 2y - 4x - \frac{1}{2}y - 2 - 1 = 0 \]

This simplifies to:

\[ -x + \frac{3}{2}y - 3 = 0 \]

Rearranging further, we get:

\[ x - \frac{3}{2}y = -3 \quad \text{(Equation 2)} \]

### Step 5: Solve the system of equations

Now we have two equations:

1. \( 2x - y = 2 \) (Equation 1)
2. \( x - \frac{3}{2}y = -3 \) (Equation 2)

We can solve these equations simultaneously. From Equation 1, we can express \( y \) in terms of \( x \):

\[ y = 2x - 2 \]

Substituting this expression for \( y \) into Equation 2:

\[ x - \frac{3}{2}(2x - 2) = -3 \]

### Step 6: Simplify and solve for \( x \)

Expanding the equation:

\[ x - 3x + 3 = -3 \]

Combining like terms:

\[ -2x + 3 = -3 \]

Subtracting 3 from both sides:

\[ -2x = -6 \]

Dividing by -2:

\[ x = 3 \]

### Step 7: Substitute \( x \) back to find \( y \)

Now substitute \( x = 3 \) back into the equation for \( y \):

\[ y = 2(3) - 2 = 6 - 2 = 4 \]

### Step 8: Find the length of the sides

Now we can find the length of the sides by substituting \( x \) and \( y \) back into any of the original side equations. Let's use \( a = x + 3y \):

\[ a = 3 + 3(4) = 3 + 12 = 15 \]

Thus, the length of each side of the equilateral triangle is \( 15 \).

### Summary of the Solution

The sides of the equilateral triangle are equal and each side measures \( 15 \).

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