To solve the matrix equation
[tex]\[
\left[\begin{array}{cc}
3 & 7 \\
2 & 5
\end{array}\right]
\left[\begin{array}{c}
x \\
y
\end{array}\right]
=
\left[\begin{array}{c}
14 \\
10
\end{array}\right],
\][/tex]
we need to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy the system of equations represented by the matrix multiplication.
Let's write out the system of linear equations represented by this matrix equation:
[tex]\[
\begin{cases}
3x + 7y = 14 \\
2x + 5y = 10
\end{cases}
\][/tex]
We will use the method of elimination to solve this system.
1. Multiply the first equation by 2 and the second equation by 3 to align the coefficients of [tex]\( x \)[/tex]:
[tex]\[
\begin{cases}
2(3x + 7y) = 2(14) \implies 6x + 14y = 28 \\
3(2x + 5y) = 3(10) \implies 6x + 15y = 30
\end{cases}
\][/tex]
2. Subtract the first modified equation from the second modified equation to eliminate [tex]\( x \)[/tex]:
[tex]\[
(6x + 15y) - (6x + 14y) = 30 - 28 \implies y = 2
\][/tex]
3. Now that we have [tex]\( y = 2 \)[/tex], substitute it back into one of the original equations (we'll use [tex]\( 3x + 7y = 14 \)[/tex]) to find [tex]\( x \)[/tex]:
[tex]\[
3x + 7(2) = 14 \implies 3x + 14 = 14 \implies 3x = 0 \implies x = 0
\][/tex]
Therefore, the solution to the matrix equation is [tex]\( (x, y) = (0, 2) \)[/tex].
Thus, the correct solution is:
[tex]\[
(0.0, 2.0)
\][/tex]
