Question

Use the matrix tool to solve the system of equations. Choose the correct ordered pair.7x - 4y = 13 2x - 5y = -4

Answer 1

Solutions

In Matrix we use initially based on systems of linear equations.The matrix method is similar to the method of Elimination as but is a lot cleaner than the elimination method.Solving systems of equations by Matrix Method involves expressing the system of equations in form of a matrix and then reducing that matrix into what is known as Row Echelon Form.

Calculations

⇒ Rewrite the linear equations above as a matrix


`\left[\begin{array}{ccc}7&-4&13\\2&-5&-4\\\end{array}\right]`

⇒ Apply to Row₂ : Row₂ - 2/7 Row₁


`\left[\begin{array}{ccc}7&-4&13\\0&-27/7&-54/7\\\end{array}\right]`

⇒ Simplify rows


`\left[\begin{array}{ccc}7&-4&13\\0&1&2\\\end{array}\right]`

Note: The matrix is now in echelon form.
The steps below are for back substitution.

⇒ Apply to Row₁ : Row₁ + 4 Row₂


`\left[\begin{array}{ccc}7&0&21\\0&1&2\\\end{array}\right]`

⇒ Simplify rows


`\left[\begin{array}{ccc}1&0&3\\0&1&2\\\end{array}\right]`

⇒ Therefore,


`x=3 y=2`

Answer 2

`\left\{\begin{array}{ccc}7x-4y=13\\2x-5y=-4\end{array}\right\\\begin{array}{ccc}(i)\\(ii)\end{array}\left[\begin{array}{ccc}7&-4\\2&-5\end{array}\right\left|\begin{array}{ccc}13\\-4\end{array}\right]\xrightarrow{(i):7}\left[\begin{array}{ccc}1&-(4)/(7)\\2&-5\end{array}\right\left|\begin{array}{ccc}(13)/(7)\\-4\end{array}\right]\xrightarrow{(ii)-2(i)}`

`\left[\begin{array}{ccc}1&-(4)/(7)\\0&-(27)/(7)\end{array}\right\left|\begin{array}{ccc}(13)/(7)\\-(54)/(7)\end{array}\right]\xrightarrow{(ii)\cdot(-(7)/(27))}\\\\\left[\begin{array}{ccc}1&-(4)/(7)\\0&1\end{array}\right\left|\begin{array}{ccc}(13)/(7)\\2\end{array}\right]\xrightarrow{(i)+(ii)\cdot(4)/(7)}`

`\left[\begin{array}{ccc}1&0\\0&1\end{array}\right\left|\begin{array}{ccc}3\\2\end{array}\right]\\\\Answer:\boxed{ \left \{ {{x=3} \atop {y=2}} \right. }\to\fbox{B.}`