For the system of linear equations 2 x+4 y+2 a z=b x+2 y+3 z=4 2 x-5 y+2 z=8 which of the …

Question

For the system of linear equations
$2 x+4 y+2 a z=b$
$x+2 y+3 z=4$
$2 x-5 y+2 z=8$
which of the following is NOT correct?

Accepted Answer

The given system of equations is :

1. $2x + 4y + 2az = b$

2. $x + 2y + 3z = 4$

3. $2x - 5y + 2z = 8$

We can write this in matrix form :

$$
\begin{bmatrix}
2 & 4 & 2a \
1 & 2 & 3 \
2 & -5 & 2
\end{bmatrix}
\begin{bmatrix}
x \
y \
z
\end{bmatrix}
=
\begin{bmatrix}
b \
4 \
8
\end{bmatrix}
$$

First, we need to find the determinant of the coefficient matrix, which we'll call \Delta . The coefficient matrix is :

$$
\begin{bmatrix}
2 & 4 & 2a \
1 & 2 & 3 \
2 & -5 & 2
\end{bmatrix}
$$

We find its determinant for 3 $$ \times $$ 3 matrices :

$$
\Delta = 2(2)(2) + 4(3)(2) + 2a(1)(-5) - 2a(2)(2) - 4(1)(2) - 2(3)(-5)
$$

$$
\Delta = 8 + 24 - 10a - 8a - 8 + 30
$$

$$
\Delta = -18a + 54
$$

$$
\Delta = -18(a - 3)
$$

Next, we substitute the third column of our matrix with the column of constants (b, 4, 8) and calculate the determinant \Delta \_3 :

$$
\begin{bmatrix}
2 & 4 & b \
1 & 2 & 4 \
2 & -5 & 8
\end{bmatrix}
$$

We find its determinant :

$$
\Delta_3 = 2(2)(8) + 4(4)(2) + b(1)(-5) - b(2)(2) - 4(1)(8) - 2(4)(-5)
$$

$$
\Delta_3 = 32 + 32 - 5b - 4b - 32 + 40
$$

$$
\Delta_3 = -9b + 72
$$

$$
\Delta_3 = 9(8 - b)
$$

Now, let's analyze the options :

1. For a=3 and b=8, we have \Delta  = 0 and \Delta \_3 = 0, which indicates an infinite number of solutions.

2. For a=3 and b=6, we have \Delta  = 0 and \Delta \_3 \neq  0, which indicates no solution.

3. For a=8 and b=8, we have \Delta  \neq  0, which indicates a unique solution.

4. For a=6 and b=6, we have \Delta  \neq  0, which indicates a unique solution.

Therefore, the statement that is NOT correct is Option B: "It has infinitely many solutions if a=3, b=6", because in this case the system actually has no solution.

Solutions · 1