Find the following system of equations is consistent, `(a+1)^3x+(a+2)^3y=(a+3)^3` `(a+1)x+(a+2)y=a+3` +=1, then find the value of `adot`

Explanation

To determine the value of \( a \) for which the given system of equations is consistent, we can use the concept of determinants. The system of equations is:

1. \( (a+1)^3x + (a+2)^3y = (a+3)^3 \)
2. \( (a+1)x + (a+2)y = a+3 \)

### Step 1: Write the system in standard form
We can rewrite the equations in the standard form \( Ax + By = C \):

1. \( (a+1)^3x + (a+2)^3y - (a+3)^3 = 0 \)
2. \( (a+1)x + (a+2)y - (a+3) = 0 \)

### Step 2: Set up the determinant
For the system to be consistent, the determinant of the coefficients must be zero. The determinant \( \Delta \) for the coefficients of \( x \) and \( y \) can be set up as follows:

\[ \Delta = \begin{vmatrix} (a+1)^3 & (a+2)^3 \\ (a+1) & (a+2) \end{vmatrix} \]

### Step 3: Calculate the determinant
The determinant can be calculated using the formula for a 2x2 matrix:

\[ \Delta = (a+1)^3 \cdot (a+2) - (a+2)^3 \cdot (a+1) \]

### Step 4: Simplify the determinant
Expanding the determinant:

\[ \Delta = (a+1)^3(a+2) - (a+2)^3(a+1) \]

Factor out \( (a+1)(a+2) \):

\[ \Delta = (a+1)(a+2) \left( (a+1)^2 - (a+2)^2 \right) \]

### Step 5: Simplify the expression inside the parentheses
Now simplify \( (a+1)^2 - (a+2)^2 \):

\[ (a+1)^2 - (a+2)^2 = (a^2 + 2a + 1) - (a^2 + 4a + 4) = -2a - 3 \]

### Step 6: Set the determinant to zero
For the system to be consistent, we set the determinant to zero:

\[ (a+1)(a+2)(-2a - 3) = 0 \]

### Step 7: Solve for \( a \)
This gives us three cases to solve:

1. \( a + 1 = 0 \) → \( a = -1 \)
2. \( a + 2 = 0 \) → \( a = -2 \)
3. \( -2a - 3 = 0 \) → \( a = -\frac{3}{2} \)

### Conclusion
The values of \( a \) for which the system of equations is consistent are \( a = -1 \), \( a = -2 \), and \( a = -\frac{3}{2} \). Since the question specifically asks for the value of \( a \) when \( a + 1 = 0 \), we find that:

\[ \text{The value of } a \text{ is } -2. \]