To solve for the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the given matrix equation:
[tex]\[
\begin{pmatrix}
3x + 2 & 0 \\
x + 4y & 2
\end{pmatrix}
=
\begin{pmatrix}
8 & 0 \\
2 & 2
\end{pmatrix},
\][/tex]
we need to set each corresponding element of the two matrices equal to each other. This will provide us with a system of equations to solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
### Step-by-Step Solution
1. Identify Corresponding Elements:
- From the top-left element: [tex]\( 3x + 2 = 8 \)[/tex]
- From the bottom-left element: [tex]\( x + 4y = 2 \)[/tex]
- The other corresponding elements provide [tex]\( 0 = 0 \)[/tex] and [tex]\( 2 = 2 \)[/tex], which are trivially true and do not provide additional information.
2. Solve the First Equation for [tex]\( x \)[/tex]:
Consider the equation [tex]\( 3x + 2 = 8 \)[/tex].
[tex]\[
3x + 2 = 8
\][/tex]
Subtract 2 from both sides:
[tex]\[
3x = 6
\][/tex]
Divide both sides by 3:
[tex]\[
x = 2
\][/tex]
3. Substitute [tex]\( x = 2 \)[/tex] into the Second Equation to Solve for [tex]\( y \)[/tex]:
Now use the second equation [tex]\( x + 4y = 2 \)[/tex] with [tex]\( x = 2 \)[/tex].
[tex]\[
2 + 4y = 2
\][/tex]
Subtract 2 from both sides:
[tex]\[
4y = 0
\][/tex]
Divide both sides by 4:
[tex]\[
y = 0
\][/tex]
### Summary
Thus, the values are:
[tex]\[
x = 2, \quad y = 0
\][/tex]
These values satisfy the given matrix equation.
