Question

`\left. \begin{cases} { 8 x + 2 y = 46 } \\ { 7 x + 3 y = 47 } \end{cases} \right.`

Solve for x & y using MATRICES!! Help...​

Answer 1

`\huge \boxed{\mathfrak{Question} \downarrow}`

`\left. \begin{cases} { 8 x + 2 y = 46 } \\ { 7 x + 3 y = 47 } \end{cases} \right.`

`\large \boxed{\mathfrak{Answer \: with \: Explanation} \downarrow}`

`\left. \begin{cases} { 8 x + 2 y = 46 } \\ { 7 x + 3 y = 47 } \end{cases} \right.`

First, write both the equations in its standard form.

`8x+2y=46\\ 7x+3y=47`

Now, write the equations in form of matrix.

`\left(\begin{matrix}8&2\\7&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}46\\47\end{matrix}\right)`

Then, multiply the equation towards the left by using the inverse of matrix
`\left(\begin{matrix}8&2\\7&3\end{matrix}\right)`

`\sf \: inverse(\left(\begin{matrix}8&2\\7&3\end{matrix}\right))\left(\begin{matrix}8&2\\7&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&2\\7&3\end{matrix}\right))\left(\begin{matrix}46\\47\end{matrix}\right)`

The product of the matrix & its inverse will be the identity matrix.

`\sf\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&2\\7&3\end{matrix}\right))\left(\begin{matrix}46\\47\end{matrix}\right)`

Now, multiply the matrices that lie on the left-hand side of the equal sign.

`\sf\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&2\\7&3\end{matrix}\right))\left(\begin{matrix}46\\47\end{matrix}\right)`

For the 2 × 2 matrix
`\left(\begin{matrix}a&b\\c&d\end{matrix}\right)`, the inverse matrix is ⇨
`\left(\begin{matrix}(d)/(ad-bc)&(-b)/(ad-bc)\\(-c)/(ad-bc)&(a)/(ad-bc)\end{matrix}\right)`.

`\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}(3)/(8* 3-2* 7)&-(2)/(8* 3-2* 7)\\-(7)/(8* 3-2* 7)&(8)/(8* 3-2* 7)\end{matrix}\right)\left(\begin{matrix}46\\47\end{matrix}\right)`

Do the calculations.

`\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}(3)/(10)&-(1)/(5)\\-(7)/(10)&(4)/(5)\end{matrix}\right)\left(\begin{matrix}46\\47\end{matrix}\right)`

Multiply the matrices.

`\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}(3)/(10)* 46-(1)/(5)* 47\\-(7)/(10)* 46+(4)/(5)* 47\end{matrix}\right)`

Do the arithmetics again.

`\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}(22)/(5) \\(27)/(5)\end{matrix}\right)`

Finally, extract the matrix elements x & y & write them separately.

`\large \boxed{ \boxed{ \bf \: x=(22)/(5),y=(27)/(5) }}`