Question

Solve the system by using a matrix equation

Answer 1

Answer:

Option A is correct (17,11).

Step-by-step explanation:

6x - 9y = 3

3x - 4y =7

it can be represented in matrix form as$\left[\begin{array}{cc}6&-9\\3&4\end{array}\right] \left[\begin{array}{c}x\\y\end{array}\right] = \left[\begin{array}{c}3\\7\end{array}\right]$

A= $\left[\begin{array}{cc}6&-9\\3&4\end{array}\right]$

X= $\left[\begin{array}{c}x\\y\end{array}\right]$

B= $\left[\begin{array}{c}3\\7\end{array}\right]$

i.e, AX=B

or X= A⁻¹ B

A⁻¹ = 1/|A| * Adj A

determinant of A = |A|= (6*-4) - (-9*3)

                                    = (-24)-(-27)

                                    = (-24) + 27 = 3

so, |A| = 3

Adj A=  $\left[\begin{array}{cc}-4&9\\-3&6\end{array}\right]$

A⁻¹ =  $\left[\begin{array}{cc}-4&9\\-3&6\end{array}\right]$/3

A⁻¹ =  $\left[\begin{array}{cc}-4/3&3\\-1&2\end{array}\right]$

X= A⁻¹ B

X=  $\left[\begin{array}{cc}-4/3&3\\-1&2\end{array}\right] *\left[\begin{array}{c}3\\7\end{array}\right]$

X= $\left[\begin{array}{c}(-4/3*3) + (3*7)\\(-1*3) + (2*7)\end{array}\right]$

X= $\left[\begin{array}{c}-4+21\\-3+14\end{array}\right]$

X= $\left[\begin{array}{c}17\\11\end{array}\right]$

x= 17, y= 11

solution set= (17,11).

Answer 2

Answer:

a. (17,11)

Step-by-step explanation:

The given system is ;

$6x-9y=3$

$3x-4y=7$

The augmented matrices is

$\left[\begin{array}{ccc}6&-9&|3\\3&-4&|7\end{array}\right]$

Divide Row 1 by 6

$\left[\begin{array}{ccc}1&-\frac{3}{2}&|\frac{1}{2}\\3&-4&|7\end{array}\right]$

Subtract 3 times Row 1 from Row 2

$\left[\begin{array}{ccc}1&-\frac{3}{2}&|\frac{1}{2}\\0&\frac{1}{2}&|\frac{11}{2}\end{array}\right]$

Divide Row 2 by $\frac{1}{2}$

$\left[\begin{array}{ccc}1&-\frac{3}{2}&|\frac{1}{2}\\0&1}&|11\end{array}\right]$

Add $\frac{3}{2}$ times Row 2 to Row 1

$\left[\begin{array}{ccc}1&0&|17\\0&1}&|11\end{array}\right]$

Hence the solution is (17,11)