To find the values of the variables [tex]\(a\)[/tex], [tex]\(z\)[/tex], [tex]\(m\)[/tex], and [tex]\(k\)[/tex] from the given matrix equation, let's break it down element by element.
Given matrices:
[tex]\[
\left[\begin{array}{ccc}
a+10 & 6z+1 & 6m \\
11k & 4 & 0
\end{array}\right]+\left[\begin{array}{rrr}
9a & 8z & 4m \\
11k & 2 & 2
\end{array}\right]=\left[\begin{array}{rrr}
30 & -55 & 80 \\
66 & 6 & 2
\end{array}\right]
\][/tex]
### Solving for [tex]\( a \)[/tex]:
Looking at the element in the first row and first column:
[tex]\[ (a + 10) + (9a) = 30 \][/tex]
Combine like terms:
[tex]\[ a + 10 + 9a = 30 \][/tex]
[tex]\[ 10a + 10 = 30 \][/tex]
Subtract 10 from both sides:
[tex]\[ 10a = 20 \][/tex]
Divide by 10:
[tex]\[ a = 2 \][/tex]
### Solving for [tex]\( z \)[/tex]:
Looking at the element in the first row and second column:
[tex]\[ (6z + 1) + (8z) = -55 \][/tex]
Combine like terms:
[tex]\[ 6z + 1 + 8z = -55 \][/tex]
[tex]\[ 14z + 1 = -55 \][/tex]
Subtract 1 from both sides:
[tex]\[ 14z = -56 \][/tex]
Divide by 14:
[tex]\[ z = -4 \][/tex]
### Solving for [tex]\( m \)[/tex]:
Looking at the element in the first row and third column:
[tex]\[ (6m) + (4m) = 80 \][/tex]
Combine like terms:
[tex]\[ 6m + 4m = 80 \][/tex]
[tex]\[ 10m = 80 \][/tex]
Divide by 10:
[tex]\[ m = 8 \][/tex]
### Solving for [tex]\( k \)[/tex]:
Looking at the element in the second row and first column:
[tex]\[ (11k) + (11k) = 66 \][/tex]
Combine like terms:
[tex]\[ 11k + 11k = 66 \][/tex]
[tex]\[ 22k = 66 \][/tex]
Divide by 22:
[tex]\[ k = 3 \][/tex]
So the values of the variables are:
[tex]\[ a = 2 \][/tex]
[tex]\[ z = -4 \][/tex]
[tex]\[ m = 8 \][/tex]
[tex]\[ k = 3 \][/tex]
Therefore:
[tex]\[ a = 2 \][/tex]
[tex]\[ z = -4 \][/tex]
[tex]\[ m = 8 \][/tex]
[tex]\[ k = 3 \][/tex]
