To solve the matrix equation
[tex]\[
\begin{bmatrix}
1 & 2 \\
3 & 7
\end{bmatrix}
\begin{bmatrix}
x \\
y
\end{bmatrix} =
\begin{bmatrix}
10 \\
21
\end{bmatrix},
\][/tex]
we need to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy this equation. Here's a step-by-step approach:
1. Write the system of linear equations: The given matrix equation corresponds to the following system of linear equations:
[tex]\[
1x + 2y = 10
\][/tex]
[tex]\[
3x + 7y = 21
\][/tex]
2. Solve one of the equations for one variable: Let's solve the first equation for [tex]\(x\)[/tex]:
[tex]\[
x + 2y = 10 \Rightarrow x = 10 - 2y
\][/tex]
3. Substitute this expression into the second equation:
[tex]\[
3(10 - 2y) + 7y = 21
\][/tex]
Simplify this equation:
[tex]\[
30 - 6y + 7y = 21 \Rightarrow 30 + y = 21
\][/tex]
4. Solve for [tex]\(y\)[/tex]:
[tex]\[
30 + y = 21 \Rightarrow y = 21 - 30 \Rightarrow y = -9
\][/tex]
5. Substitute [tex]\(y\)[/tex] back into the expression for [tex]\(x\)[/tex]:
[tex]\[
x = 10 - 2y = 10 - 2(-9) = 10 + 18 = 28
\][/tex]
Thus, the solution to the matrix equation is:
[tex]\[
\begin{bmatrix}
x \\
y
\end{bmatrix} =
\begin{bmatrix}
28 \\
-9
\end{bmatrix}
\][/tex]
So, the correct choice is [tex]\((28, -9)\)[/tex]. Therefore, the answer is (28, -9).
