Answer:
[tex]x=2[/tex]
Step-by-step explanation:
The given matrix equation is ;
[tex]\frac{3}{2}\left[\begin{array}{cc}x&6\\8&4\end{array}\right] +y \left[\begin{array}{cc}1&4\\3&2\end{array}\right] =\left[\begin{array}{cc}z&z\\6z&2\end{array}\right][/tex]
We multiply the scalars to get;
[tex]\left[\begin{array}{cc}\frac{3}{2}x&9\\12&6\end{array}\right] +\left[\begin{array}{cc}y&4y\\3y&2y\end{array}\right] =\left[\begin{array}{cc}z&z\\6z&2\end{array}\right][/tex]
We simplify to get;
[tex]\left[\begin{array}{cc}\frac{3}{2}x+y&9+4y\\12+3y&6+2y\end{array}\right] =\left[\begin{array}{cc}z&z\\6z&2\end{array}\right][/tex]
By the equality property of matrices, we can write the following equations;
[tex]\frac{3}{2}x+y=z...(1)[/tex]
[tex]6+2y=2[/tex]
[tex]2y=2-6[/tex]
[tex]2y=-4[/tex]
[tex]y=-2[/tex]
Also;
[tex]12+3y=6z[/tex]
[tex]\Rightarrow 12+3(-2)=6z[/tex]
[tex]\Rightarrow 12-6=6z[/tex]
[tex]\Rightarrow 6=6z[/tex]
[tex]\Rightarrow z=1[/tex]
We substitute these values into equation (1) to find x.
[tex]\frac{3}{2}x-2=1[/tex]
[tex]\frac{3}{2}x=1+2[/tex]
[tex]\frac{3}{2}x=3[/tex]
[tex]\frac{1}{2}x=1[/tex]
[tex]x=2[/tex]
