Find the values of the variables for which the statement is true, if possible.

[tex]\[
\begin{bmatrix}
-7 & -7 & x \\
0 & 6 & y+4 \\
-6 & -5 & z
\end{bmatrix}
=
\begin{bmatrix}
-7 & m+4 & 4 \\
0 & 6 & 5 \\
-6 & -5 & -1
\end{bmatrix}
\][/tex]

The statement can be true if [tex]\( m = \ \square \)[/tex], [tex]\( x = \ \square \)[/tex], [tex]\( y = \ \square \)[/tex], and [tex]\( z = \ \square \)[/tex].



Answer :

We need to find the values of the variables [tex]\( m \)[/tex], [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] such that the given matrices are equal. Let's compare corresponding elements from both matrices step-by-step.

We start with the first row:
[tex]\[ \begin{array}{ccc} -7 & -7 & x \\ \end{array} \][/tex]
must equal
[tex]\[ \begin{array}{ccc} -7 & m+4 & 4 \\ \end{array} \][/tex]

From the first element:
[tex]\[ -7 = -7 \][/tex]
This equation is already satisfied; no variable here.

From the second element:
[tex]\[ -7 = m + 4 \][/tex]
Solving for [tex]\( m \)[/tex]:
[tex]\[ m = -7 - 4 \implies m = -11 \][/tex]

From the third element:
[tex]\[ x = 4 \][/tex]

Now, let's move to the second row:
[tex]\[ \begin{array}{ccc} 0 & 6 & y+4 \\ \end{array} \][/tex]
must equal
[tex]\[ \begin{array}{ccc} 0 & 6 & 5 \\ \end{array} \][/tex]

From the first element:
[tex]\[ 0 = 0 \][/tex]
This equation is already satisfied; no variable here.

From the second element:
[tex]\[ 6 = 6 \][/tex]
This equation is already satisfied; no variable here.

From the third element:
[tex]\[ y + 4 = 5 \][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[ y = 5 - 4 \implies y = 1 \][/tex]

Finally, let's look at the third row:
[tex]\[ \begin{array}{ccc} -6 & -5 & z \\ \end{array} \][/tex]
must equal
[tex]\[ \begin{array}{ccc} -6 & -5 & -1 \\ \end{array} \][/tex]

From the first element:
[tex]\[ -6 = -6 \][/tex]
This equation is already satisfied; no variable here.

From the second element:
[tex]\[ -5 = -5 \][/tex]
This equation is already satisfied; no variable here.

From the third element:
[tex]\[ z = -1 \][/tex]

Therefore, the values that satisfy the given matrix equation are:
[tex]\[ m = -11, \quad x = 4, \quad y = 1, \quad z = -1 \][/tex]

Thus, the statement can be true if [tex]\( m = -11 \)[/tex], [tex]\( x = 4 \)[/tex], [tex]\( y = 1 \)[/tex], and [tex]\( z = -1 \)[/tex].