To find the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] given the matrix equality
[tex]\[
\begin{pmatrix}
a + 2 & b - 4 \\
1 & -2 + c
\end{pmatrix}
=
\begin{pmatrix}
3a - 2 & 5b + 6 \\
1 & 4 + 2c
\end{pmatrix},
\][/tex]
we shall equate corresponding elements of the two matrices and solve the resulting system of equations.
### Step 1: Equating corresponding elements
1. For the element at the position [tex]\((1,1)\)[/tex], we equate:
[tex]\[
a + 2 = 3a - 2
\][/tex]
2. For the element at the position [tex]\((1,2)\)[/tex], we equate:
[tex]\[
b - 4 = 5b + 6
\][/tex]
3. For the element at the position [tex]\((2,2)\)[/tex], we equate:
[tex]\[
-2 + c = 4 + 2c
\][/tex]
(Note: The element at position [tex]\((2,1)\)[/tex] is [tex]\(1\)[/tex] in both matrices, so it does not provide additional information.)
### Step 2: Solving the equations
Equation 1:
[tex]\[
a + 2 = 3a - 2
\][/tex]
Rearrange to solve for [tex]\(a\)[/tex]:
[tex]\[
a + 2 = 3a - 2 \implies 2 + 2 = 3a - a \implies 4 = 2a \implies a = 2
\][/tex]
Equation 2:
[tex]\[
b - 4 = 5b + 6
\][/tex]
Rearrange to solve for [tex]\(b\)[/tex]:
[tex]\[
b - 4 = 5b + 6 \implies -4 - 6 = 5b - b \implies -10 = 4b \implies b = -\frac{10}{4} \implies b = -\frac{5}{2}
\][/tex]
Equation 3:
[tex]\[
-2 + c = 4 + 2c
\][/tex]
Rearrange to solve for [tex]\(c\)[/tex]:
[tex]\[
-2 + c = 4 + 2c \implies -2 - 4 = 2c - c \implies -6 = c \implies c = -6
\][/tex]
### Conclusion
The values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are:
[tex]\[
a = 2, \quad b = -\frac{5}{2}, \quad c = -6
\][/tex]
