if the system of following equations in x and y
`2x+3y+4=0,3x+5y+6=0`
`2x^(2)+6xy+5y^(2)+8x+12y+1=t`
Consists of a solution, then find the value of `|t|`

Explanation

To solve the problem, we need to determine the value of \( |t| \) such that the given system of equations has a solution. The equations are:

1. \( 2x + 3y + 4 = 0 \) (Equation 1)
2. \( 3x + 5y + 6 = 0 \) (Equation 2)
3. \( 2x^2 + 6xy + 5y^2 + 8x + 12y + 1 = t \) (Equation 3)

### Step 1: Rewrite the equations
We can express Equations 1 and 2 in a standard form:

- From Equation 1: \( 2x + 3y = -4 \)
- From Equation 2: \( 3x + 5y = -6 \)

### Step 2: Form the coefficient matrix
The system of equations can be represented in matrix form as follows:

\[ \begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} -4 \\ -6 \end{bmatrix} \]

### Step 3: Check for consistency using determinants
To check if the system has a solution, we can calculate the determinant of the coefficient matrix:

\[ D = \begin{vmatrix} 2 & 3 \\ 3 & 5 \end{vmatrix} = (2)(5) - (3)(3) = 10 - 9 = 1 \]

Since \( D \neq 0 \), the system of equations is consistent and has a unique solution.

### Step 4: Substitute the solution into Equation 3
Next, we need to find the values of \( x \) and \( y \) that satisfy both equations. We can solve the system using substitution or elimination.

Let's use elimination. Multiply Equation 1 by 3 and Equation 2 by 2:

\[ 6x + 9y = -12 \quad \text{(Equation 1 multiplied by 3)} \]
\[ 6x + 10y = -12 \quad \text{(Equation 2 multiplied by 2)} \]

Now subtract the first modified equation from the second:

\[ (6x + 10y) - (6x + 9y) = -12 + 12 \]
\[ y = 0 \]

Substituting \( y = 0 \) back into Equation 1:

\[ 2x + 3(0) + 4 = 0 \implies 2x + 4 = 0 \implies 2x = -4 \implies x = -2 \]

Thus, we have \( x = -2 \) and \( y = 0 \).

### Step 5: Substitute \( x \) and \( y \) into Equation 3
Now we substitute \( x = -2 \) and \( y = 0 \) into Equation 3:

\[ 2(-2)^2 + 6(-2)(0) + 5(0)^2 + 8(-2) + 12(0) + 1 = t \]

Calculating this gives:

\[ 2(4) + 0 + 0 - 16 + 0 + 1 = t \]
\[ 8 - 16 + 1 = t \implies t = -7 \]

### Step 6: Find \( |t| \)
Finally, we find the absolute value of \( t \):

\[ |t| = |-7| = 7 \]

### Final Answer
The value of \( |t| \) is \( \boxed{7} \).